Something is intuitively obvious if all mathematicians, most of them, some of them or the author would expect that something to be true. For instance, it is intuitively obvious that pi is normal and that P!=NP. It is probably not intuitively obvious that Fermat's last theorem is true (nor is it obvious that it is false).
Proving that something which is intuitively obvious is in fact true might be hard; it may even be impossible for us (although for obvious reasons we have no examples of this). The depth of the intuition required varies tremendously; some things termed intuitively obvious aren't.
For obvious reasons, the phrase is somewhat notorious, and not liked by all.