What does it mean?
It means that you should not try to use the same iterator in two threads. If you have two threads that need to iterate over the keys, values or entries, then they each should create and use their own iterators.
What happens if I try to iterate the map with two threads at the same time?
It is not entirely clear what would happen if you broke this rule. You could just get confusing behavior, in the same way that you do if (for example) two threads try to read from standard input without synchronizing. You could also get non-thread-safe behavior.
But if the two threads used different iterators, you should be fine.
What happens if I put or remove a value from the map while iterating it?
That's a separate issue, but the javadoc section that you quoted adequately answers it. Basically, the iterators are thread-safe, but it is not defined whether you will see the effects of any concurrent insertions, updates or deletions reflected in the sequence of objects returned by the iterator. In practice, it probably depends on where in the map the updates occur.