The New Advent page for John 15 can be found here. You can listen to the Latin here and the Greek here.
Ego sum vitis vera, et Pater meus agricola est. 2 Omnem palmitem in me non ferentem fructum, tollet eum, et omnem qui fert fructum, purgabit eum, ut fructum plus afferat. 3 Jam vos mundi estis propter sermonem quem locutus sum vobis. 4 Manete in me, et ego in vobis. Sicut palmes non potest fere fructum a semetipso, nisi manserit in vite, sic nec vos, nisi in me manseritis. 5 Ego sum vitis, vos palmites: qui manet in me, et ego in eo, hic fert fructum multum, quia sine me nihil potestis facere. 6 Si quis in me non manserit, mittetur foras sicut palmes, et arescet, et colligent eum, et in ignem mittent, et ardet. 7 Si manseritis in me, et verba mea in vobis manserint, quodcumque volueritis petetis, et fiet vobis. 8 In hoc clarificatus est Pater meus, ut fructum plurimum afferatis, et efficiamini mei discipuli. 9 Sicut dilexit me Pater, et ego dilexi vos. Manete in dilectione mea. 10 Si præcepta mea servaveritis, manebitis in dilectione mea, sicut et ego Patris mei præcepta servavi, et maneo in ejus dilectione.11 Hæc locutus sum vobis: ut gaudium meum in vobis sit, et gaudium vestrum impleatur. 12 Hoc est præceptum meum, ut diligatis invicem, sicut dilexi vos. 13 Majorem hac dilectionem nemo habet, ut animam suam ponat qui pro amicis suis. 14 Vos amici mei estis, si feceritis quæ ego præcipio vobis. 15 Jam non dicam vos servos: quia servus nescit quid faciat dominus ejus. Vos autem dixi amicos: quia omnia quæcumque audivi a Patre meo, nota feci vobis. 16 Non vos me elegistis, sed ego elegi vos, et posui vos ut eatis, et fructum afferatis, et fructus vester maneat: ut quodcumque petieritis Patrem in nomine meo, det vobis. .
And the Douay-Rheims translation:
I am the true vine; and my Father is the husbandman.  Every branch in me, that beareth not fruit, he will take away: and every one that beareth fruit, he will purge it, that it may bring forth more fruit.  Now you are clean by reason of the word, which I have spoken to you.  Abide in me, and I in you. As the branch cannot bear fruit of itself, unless it abide in the vine, so neither can you, unless you abide in me.  I am the vine: you the branches: he that abideth in me, and I in him, the same beareth much fruit: for without me you can do nothing. If any one abide not in me, he shall be cast forth as a branch, and shall wither, and they shall gather him up, and cast him into the fire, and he burneth.  If you abide in me, and my words abide in you, you shall ask whatever you will, and it shall be done unto you.  In this is my Father glorified; that you bring forth very much fruit, and become my disciples.  As the Father hath loved me, I also have loved you. Abide in my love.  If you keep my commandments, you shall abide in my love; as I also have kept my Father' s commandments, and do abide in his love. These things I have spoken to you, that my joy may be in you, and your joy may be filled.  This is my commandment, that you love one another, as I have loved you.  Greater love than this no man hath, that a man lay down his life for his friends.  You are my friends, if you do the things that I command you.  I will not now call you servants: for the servant knoweth not what his lord doth. But I have called you friends: because all things whatsoever I have heard of my Father, I have made known to you. You have not chosen me: but I have chosen you; and have appointed you, that you should go, and should bring forth fruit; and your fruit should remain: that whatsoever you shall ask of the Father in my name, he may give it you. .
Catena Aurea on the image of Christ as the vine:
HILARY. He rises in haste to perform the sacrament of His final passion in the flesh (such is His desire to fulfill His Father's commandment) and therefore takes occasion to unfold the mystery of His assumption of His flesh, whereby He supports us, as the vine does its branches: I am the true vine.
AUG. He says this as being the Head of the Church, of which we are the members, the Man Christ Jesus; for the vine and the branches are of the same nature. When He says, I am the true vine, He does not mean really a vine; for He is only called so metaphorically, not literally, even as He is called the Lamb, the Sheep, and the like; but He distinguishes Himself from that vine to whom it is said, How you are turned into the degenerate plant of a strange vine to me (Jer 11:21). For how is that a true vine, which when grapes are expected from it, produces only thorns?
HILARY. But He wholly separates this humiliation in the flesh from the form of the Paternal Majesty, by setting forth the Father as the diligent husbandman of this vine: And My Father is the husbandman.
AUG. For we cultivate God, and God cultivates us. But our culture of God does not make Him better: our culture is that of adoration, not of plowing: His culture of us makes us better. His culture consists in extirpating all the seeds of wickedness from our hearts, in opening our heart to the plow, as it were, of His word, in sowing in us the seeds of His commandments, in waiting for the fruits of piety.
The test of being on the vine is fruitfulness:
CHRYS. And forasmuch as Christ was sufficient for Himself, but His disciples needed the help of the Husbandman, of the vine He says nothing, but adds concerning the branches, Every branch in Me that bears not fruit, He takes away. By fruit is meant life, i.e. that no one can be in Him without good works.
HILARY. The useless and deceitful branches He cuts down for burning.
CHRYS. And inasmuch as even the best of men require the work of the husbandman, He adds, And every branch that bears fruit, He purges it, that it may bring forth more fruit. He alludes here to the tribulations and trials which were coming upon them, the effect of which would be to purge, and so to strengthen them. By pruning the branches we make the tree shoot out the more.
AUG. And who is there in this world so clean, that he cannot be more and more changed? Here, if we say that we have no sin, we deceive ourselves. He cleans then the clean, i.e. the fruitful, that the cleaner they be, the more fruitful they may be. Christ is the vine, in that He said, My Father is greater than I; but in that He said, I and My Father are one, He is the husbandman; not like those who carry on an external ministry only; for He gives increase within.
Thus He calls Himself immediately the cleanser of the branches: Now you are clean through the word, which I have spoken to you. He performs the part of the husbandman then, as well as of the vine. But why does He not say, you are clean by reason of the baptism wherewith you are washed? Because it is the word in the water which cleans. Take away the word, and what is the water, which but water.
Add the word to the element, and you have a sacrament. Whence has the water such virtue as that by touching the body, it cleans the heart, but by the power of the word, not spoken only, but believed? For in the word itself the passing sound is one thing, the abiding virtue another. This word of faith is of such avail in the Church of God that by Him who believes, presents, blesses, sprinkles the infant, it cleanses that infant, though itself is unable to believe.
Love is not separate from the commandments but flows from them:
CHRYS. If then I love you, be of good cheer; if it is the Father's glory that you bring forth good fruit, bear no evil. Then to rouse them to exertion, He adds, Continue you in My love; and then shows how this is to be done: If you keep My commandments, you shall abide in My love.
AUG. Who doubts that love precedes the observance of the commandments? For who loves not, has not that whereby to keep the commandments. These words then do not declare whence love arises, but how it is shown, that no one might deceive himself into thinking that he loved our Lord, when he did not keep His commandments. Though the words, Continue you in My love, do not of themselves make it evident which love He means, ours to Him, or His to us, yet the preceding words do: I love you, He says: and then immediately after, Continue you in My love.
Continue you in My love, then, is, continue in My grace; and, If you keep My commandments, you shall abide in My love, is, Your keeping of My commandments will be evidence to you that you abide in My love. It is not that we keep His commandments first, and that then He loves; but that He loves us, and then we keep His commandments. This is that grace, which is revealed to the humble, but hidden from the proud. But what means the next words, Even as I have kept My Father's commandments, and abide in His love: i.e., the Father's love, wherewith He loves the Son.
Must this grace, wherewith the Father loves the Son, be understood to be like the grace wherewith the Son loves us? No; for whereas we are sons not by nature, but by grace, the Only Begotten is Son not by grace, but by nature. We must understand this then to refer to the manhood in the Son, even as the words themselves imply: As My Father has loved Me, even so love I you.
The grace of a Mediator is expressed here; and Christ is Mediator between God and man, not as God, but as man. This then we may say, that since human nature does not pertain to the nature of God, but does by grace pertain to the Person of the Son, grace also pertains to that Person: such grace as has nothing superior, nothing equal to it. For no merits on man's part preceded the assumption of that nature.
ALCUIN. Even as 1 have kept My Father's commandments. The Apostle explains what these commandments were: Christ became obedient to death, even the death of the cross (Phil 2:8).
CHRYS. Then because the Passion was now approaching to interrupt their joy, He adds, These things have I spoken to you, that my joy may remain in you: as if He said, And if sorrow fall upon you, I will take it away, so that you shall rejoice in the end.
AUG. And what is Christ's joy in us, but that He deigns to rejoice on our account? And what is our joy, which He says shall be full, but to have fellowship with Him? He had perfect joy on our account, when He rejoiced in foreknowing, and predestinating us; but that joy was not in us, because then we did not exist: it began to be in us, when He called us. And this joy we rightly call our own, this joy wherewith we shall be blessed; which is begun in the faith of them who are born again, and shall be fulfilled in the reward of them who rise again.