What is interior monologue?
Basically, an interior monologue is an author's way to let the audience see what is going on in the mind of a character. It allows us to glimpse the inner lives of characters.
The author uses stream of consciousness writing to allow us, the reader, to experience thoughts, emotions, questions, and conflicts as they arise in the life and mind of a character. It is often disjointed or fragmented and can be challenging to follow at times. It's also important to recognize that this information is presented without involvement of the narrator; it is a clear, straightforward window into the mind of the character without any prejudice or opinion.
The best example that I can think of occurs in James Joyce's novel Ulysses. Take a look at the last chapter for a really solid example of the stream of consciousness writing that an interior monologue can be.