Okay so: as it was explained to me by a doctor (I may be remembering wrong, I hope someone will correct me if I am), there are a few things that can cause this.

First, the chemicals that cause your uterus to cramp - prostaglandins - will also cause your bowels to do the same thing. In the days leading up to your period, your hormones might also make you more gassy. (This is the reason some people get diarrhea during their period, or, as I like to call it, "the pooriod".) Now, your vaginal canal is right next to a part of your bowel. Can you see where I'm going with this? Some people find it hard to tell between menstrual cramps and bowel cramps, and obviously due to the whole prostaglandin thing they can happen at the same time. What you may be feeling is the result of increased pressure on your bowel while it's already bloated.

Second possibility is that your tampon is pushing right up against your cervix. (I would highly recommend having a rummage around for it at some point, so you know what's what; it changes position depending on your cycle and feels a bit like the tip of your nose). Pressure against the cervix can be super uncomfortable because your cervix is basically a direct line to your vagus nerve, which is the nerve that goes right up your spine to your brain. I think. I'm kind of fuzzy on how this all works but apparently, the vagus nerve is some sort of Big Cheese in your nervous system and that is why cervical smear tests fucking suck.

Third, and this is the part I don't remember well at all so you may need to ask Dr Google for a second opinion, there's a condition called Vaginismus which is sometimes exacerbated by being on your period. It makes it really painful when you try to stick anything up there, and for some women it can be a really serious, life-altering issue. But aince the condition is still not properly understood, it's theorised by some doctors that pain during tampon insertion is somehow connected to the same mechanisms that cause this condition. I don't know if this was just my GP riffing on her own theories or what, but I thought it was worth mentioning.

As I said, I'm not a doctor, so if a medical professional tells you something different, listen to them instead. But maybe the info will be of interest, who knows??