In Slaughterhouse-Five, is Billy Pilgrim really time traveling? How do you know this?
The answer depends on what you mean by "really time traveling." Billy does become “unstuck” in time, but it is not clear what that means. In a sense, as other answers have pointed out, the question is moot since the Tralfamadorian experience of time does not include past or future – events happen in a constant present. Instead, think of the time traveling as Vonnegut’s attempt to replicate in his novel what a Tralfamadorian novel might be:
…each clump of-symbols is a brief, urgent message describing a situation, a scene. We Tralfamadorians read them all at once, not one after the other. There isn't any particular relationship between all the messages, except that the author has chosen them carefully, so that, when seen all at once, they produce an image of life that is beautiful and surprising and deep. There is no beginning, no middle, no end, no suspense, no moral, no causes, no effects. What we love in our books are the depths of many marvelous moments seen all at one time.
That is, Slaughterhouse Five can be seen as an attempt, within the confines of the English language, to represent the “many marvelous moments” Billy experiences all at once. This solves a practical narrative problem for Vonnegut – the jumping forwards and backwards in time allows him to juxtapose key events in Billy’s life – but also suggests that understanding what happened at Dresden requires a new kind of narrative, one that can show the interrelatedness of past, present and future. I don’t know if Billy is “really” time traveling, but Vonnegut, given he way he has structured his narrative, clearly is trying!
The question is somewhat misleading because part of the explanation for jumping back and forth in time was the fact that, particularly from the perspective of the Tralfamadorians that there is no such thing as time, what has happened and what will happen is part of an all-encompassing present, such that he might jump around, but he is not actually travelling from one time to another, just being aware at different points in the continuing motion of it.
One could argue that he is not, since his actual presence and then disappearance at any particular time would certainly cause disruptions, and there never appear to be incidents where he suddenly shows up without having been present before. So at least in my interpretation, he is not necessarily traveling, just having his awareness moved to different points in his life as it is continuously happening, a la the Tralfamadorian perspective.