Roughly the immune system says "OK, let's attack things this shape" - which is why it's so useful for defending us against a wide variety of viruses - but alas, the shape it's picking in some patients happens to also match Platelet Factor 4, so now they are mounting an immune defence against their own blood clotting mechanism which is bad.
I don't know (perhaps if I read more papers about this I'd find out) whether it is suspected that the immune system makes this bad matching choice based on the spike protein (which we want it to match because of this coronavirus killing millions of people) or the adenovirus (which even if it could replicate in humans, which it can't, basically gives you the common cold so who cares) which is delivering it. If it's the latter, that's showing an advantage to the mRNA vaccines, since they don't need a viral delivery mechanism at all.
As I understand, the required teratogenicity study in rats with Pfizer is ongoing as of two weeks ago [1]. It’s rather optimistic to claim we’ve seen everything when even animal studies haven’t completed. Vaccine science itself is rapidly developing. Now I don’t think there will be issues with this drug, but surprises may lurk still.
[1]: https://www.gov.uk/government/publications/regulatory-approv...
Vaccines are behaving differently from drugs. Once they took effect, and your immune system reacted, it is over. And with millions of people vaccinated by now, for months, any adverse effects would have shown. That's why we discuss blood clots, under normal circumstances these rare cases would have gone unnoticed for quite a while.