There's not really much difference between these sentences -- not when it comes to the information they actually convey, anyway. Today is my birthday, whether I say it happens to be, or just that it is.
Which brings us to the question of why a structure like this exists. Well, because language is an imprecise and constantly evolving tool and sometimes we have more than one way to say the same thing. No two phrases are exactly equal, though, and changes in wording can "feel" different even when they mean the same thing. The addition of "happen"/"happens" in this sort of sentence implies some sort of coincidence. I happen to have exactly what you need, not because I knew you would need it, but because I always carry it with me, or got it for some other reason, but by luck I have it here now.
However, this phrase is so often used tongue-in-cheek that it's often used when there is no coincidence at all. It has a much friendlier sound to it. Sometimes being too direct with your wording can come across as a little aggressive, and this particular construction is a common choice for how to soften it up a little.