Why does my hard palate get cut and swell every time I have toasted bagel?

It depends. If you are putting cream cheese on it, you might be allergic. If it is really hot when you are biting into it, you are burning your hard palate. If you are waiting a few minutes and the bagel is really hard, it might be so hard that it is cutting your palate. If it is repeatedly occurring it is pretty strange and unusual.
It happens. Hard or crispy foods can cut into the palate, gums, or throat. The natural response of the body is to swell after injury.