What does it mean to be "worth someone's keep"?
Solution 1:
Keep here refers to board and lodging; subsistence; support (e.g. to work for one's keep).
Worth one's keep means one is worth the trouble of lodging and feeding him. Here the disciples are traveling, and Jesus is telling them not to provide for their own boarding; that their work will be worth board and lodging to those they come to. As Kit points out below, he is also reminding them that to be worth their keep, they must work. Conversely, they shouldn't expect free handouts if they haven't done anything.