'Animal digest' - A material which results from chemical and/or enzymatic hydrolysis of clean and undecomposed animal tissue. The animal tissues used shall be exclusive of hair, horns, teeth, hooves and feathers, except in such trace amounts as might occur unavoidably in good factory practice and shall be suitable for animal feed. If it bears a name descriptive of its kind or flavor, it must correspond thereto.'
What is it used for?
Quite simply, most dried pet foods would have a low palatability without the use of flavourings - that is, your pet wouldn't eat them quite as enthusiastically. Digest is a flavouring or palatability enhancer which is often sprayed onto the kibble in small amounts to make the food more appetizing. There's nothing intrinsically wrong with it as an ingredient, and as long as the rest of the ingredients together make a complete and nutritious food then it can be thought of as spraying on a little bit of gravy.
The only argument, from the standpoint of feeding a natural diet is whether the manufacturing process to create the digest (chemical or enzymatic hydrolysis) can be considered 'natural'... and if a manufacturer is wary of including digest in their ingredient declaration, even though they use it (Burns is a case in point), what does that say about their confidence in answering this question?