Presumably juicy caterpillars contain fat?
And I think that if we were not so tidy, and if there were the occasonal animal carcasses laying about in the fields, then we'd find blue tits and other fat eaters helping themselves.
I remember a sheep-keeping acquaintance (even more eccentric than I) who, when she had a sheep slaughtered would put the head in a tree in the orchard and the tits loved it. She said it was an old custom which drew the tits to the orchards so they would get the aphids etc.
I didn't know tits did that. I thought they were seed eaters. But successful species are probably omnivorous