Without knowing the full context, it's hard to say exactly what was meant. However, on its own merit, I would say that it simply means that during the course of a journey, the person crossed a state line into another state three times. There is no way of telling whether they stopped in that state, once, twice or many times or if they took a straight route but they travelled across three states.
"We went three states over" is, to me, a rather unnatural sentence. What is the context for that one?