1. If you are an Ashkenazi and are marrying an Ashkenazi, you are marrying a cousin. If you go back a mere 50 generations, we are all cousins, probably many times over. Most of us have a common ancestor going back a few centuries at most.
2. Halacha always allowed marriage of even first cousins (both cousins have a common grandparents). Canon law, which combined Jewish law and Roman law, was more restrictive since they took some Roman rules on counting relationships, added it to some Jewish law rules, and came up with a partial ban on marriage between first and second cousins (far more restrictive than either Jewish or Roman law), some of which made it into Anglo-American law though this has largely vanished. That’s where “degrees” and “once removed” comes from.
3. Marriages of even first cousins are hardly unknown in frum communities.
4. Some people have suggested that marriage between cousins raises the liklihood of birth defects since they are likely to share negative recessive traits (which only are displayed if two people marry with the same recessive traits).