Dịch - Italian-English - Vivo per te e di te.

Here "di te" states the substance, rather than the reason, of one's life.
Unfortunately I can't find a proper way to say that phrase in English.

Perhaps "from you"...?

Really? What does "to live from" mean?
It is the first time I see this expression.

Depending on the context it may have different meanings, but in this case I think what the author wants to express is that the addressee is the "source" of his living. His life is "fed" by the other person.
Am I right?

Yes you are. As always!