You most likely have a hornworm -- or more than one.

They eat the leaves and don't eat the leaf-stalks (technically petioles). They will eat blossoms and fruit as well as leaves -- and if they're in that part of the plant, the growing tip as well.

Hornworms have some of the best camouflage in the world: you can look right at them but not see them. They are active 24 hours a day. They leave distinctive droppings shaped like 1/4" dark-green blackberries. The droppings can be found under the damaged area (on leaves or on the ground).

If you see a hornworm covered with tiny white cocoons, leave him alone: his insides have been eaten by the larvae of a parasitic moth, and his eating days are over. Those wasps will go on to infect more hornworms -- and prevent them from turning into moths that will lay more hornworm eggs.

tomato hornworm damage

The plant will grow new branches and leaves. It's okay to trim off the barren petioles. The barren stems may grow new lateral branches (aka suckers).

You can dust the plant with BT (Bacillus thuringiensis kurstaki), which is available in various "caterpillar-killing" formulas. BT kurstaki kills only caterpillars (but not other larvae which look like caterpillars, such as the pine sawfly larvae which munch on my mugo pines). There are other BTs which kill other things, so make sure you buy BTk (BT kurstaki).

Note that after the caterpillar eats some BT, he will survive a day or two before dying.

I have read (but not yet verified) that horn worms will glow under a black light. If you buy a cheap handheld black light, which are about ten bucks on amazon, and go out there at night with it, they should glow in the dark for you. One horn worm will devour a big plant in a couple days. All you have to catch is one or two, and it will make a big difference.

In my experience they often come in fours. So wherever you find one, look extra hard in that vicinity for his nasty siblings.