This is what made it click for me:
Try to play the Bach Partida 2 Allemande. Here is the music: http://erato.uvt.nl/files/imglnks/us...07-BWV1004.pdf
You will notice that it's pretty much nonstop 16th notes. Where do we breathe? That's where phrasing comes in... If you try to just jump in and play the notes without thinking about phrasing, it will sound like garbage as you gasp for breaths randomly.
Listen to this Alison Balsom recording and mark where she takes breaths: Alison Balsom - Bach, Allemande from Partita II - YouTube
There you go, you just marked out the phrases. Now go play it again.
So you tried that and it sounds a bit better now. Planning out your breaths makes it seem like not just a bunch of machine gun notes. Now you need to also take note of what notes are important, what you should emphasize or not emphasize, where to add tiny rits or accels, etc.
You're not done yet. Print out another copy of the music, listen to some more recordings and use a highlighter or something to mark their phrases:
Bach Partita n. 2 in D Minor, BWV 1004, I. Allemande - YouTube
Itzhak Perlman - Bach Partita Nš 2, BWV 1004 - YouTube
It's tempting to say, "well hey, string players don't have to breathe!" But you will notice that in fact they do "breathe" and create phrases.
Now go through and mark out yourself how YOU think it should go. If you can make this piece sound good you're in great shape. It's harder than you'd think.