"Care of" (usually written "c/o") is a shortened form of "in the care of". That is, the person named is to have the letter "in his care" until he can deliver it to the intended recipient.
When the person spoken to is told to send a letter for X "(in) care of X's lawyer", that means the letter is to be mailed to the lawyer and the lawyer will see that X receives it.
There's a mournful American folk song, Down in the Valley, one verse of which runs:
Write me a letter
Send it by mail
Send it in care of
The prisoner cannot receive communications directly; letters must be sent to the prison authorities, to be passed on to him.