"They just boil out of him!" Baker means that the bluejay's words

Animals talk to each other, of course. There can be no question about that; but I suppose there are very few people who can understand them. I never knew but one man who could. I knew he could, however, because he told me so himself. He was a middle-aged, simple-hearted miner who had lived in a lonely corner of California, among the woods and mountains, a good many years, and had studied the ways of his only neighbors, the beasts and the birds, until he believed he could accurately translate any remark which they made. This was Jim Baker. According to Jim Baker, some animals have only a limited education, and some use only simple words, and scarcely ever a comparison or a flowery figure; whereas, certain other animals have a large vocabulary, a fine command of language and a ready and fluent delivery; consequently these latter talk a great deal; they like it; they are so conscious of their talent, and they enjoy “showing off.” Baker said, that after long and careful observation, he had come to the conclusion that the bluejays were the best talkers he had found among birds and beasts. Said he:

“There's more TO a bluejay than any other creature. He has got more moods, and more different kinds of feelings than other creatures; and, mind you, whatever a bluejay feels, he can put into language. And no mere commonplace language, either, but rattling, out-and-out book-talk—and bristling with metaphor, too—just bristling! And as for command of language—why YOU never see a bluejay get stuck for a word. No man ever did. They just boil out of him!