When your stomach breaks down food, it digests it. To digest is to process food so it can be absorbed into the body or information so the mind can absorb it.
If you tell your mother some bad news, like say, that you failed your math test after studying so hard, she might say, "Let me take a moment to digest that information." She's not going to swallow the paper and let her stomach do the work. Instead, she's going to take a moment to sit with the news and understand it. As a noun, a digest is a magazine that breaks down information into understandable summaries.