It gets a little complex and IANAL, but I'll give my understanding. The US has different types of law. There's the text of the US Constitution (original + amendments). There's statutory law, passed by the legislative branch aka Congress. The executive branch creates regulatory law to carry out statutory law (e.g. Congress says "regulate pollutants" and the EPA writes rules). Then the judicial branch has case law, which interprets the Constitution and statutory law. Case law builds on itself in a process called precedent in which previous decisions inform new decisions.

Roe v. Wade was a Supreme Court decision (so case law) that recognized certain rights to an abortion based on a right to privacy. In constitutional law, "right to privacy" essentially means that personal matters are none of the government's business. That is in turn heavily based on a previous case, Griswold v. Connecticut, that ruled that a Connecticut law banning contraception was unconstitutional.

Usually decisions like this have a great deal of staying power. The principle of stare decisis (Latin for let the decision stand) is that except under certain extreme circumstances, a decision should remain case law. One such exception would be Brown v. Board of Education, which overruled Plessy v. Ferguson. Plessy ruled that racial segregation was legal as long as facilities were equal, which Brown found that such segregation was instead inherently unequal.

As for why it's being overturned now? Well, the anti-abortion right has basically spent the last 50 years engineering a takeover of the judicial branch by right wing judges. The SCOTUS especially has had a massive rightward shift, helped along recently by a power play by Mitch McConnell in 2016 to block Barack Obama's SCOTUS nominee for a year and in 2020 to quickly approve the right wing Amy Coney Barrett to replace Ruth Bader Ginsburg a week before an election.

We now have a court that is incredibly unrepresentative of the country. The makeup of the entire court has centered around this one question, and now the right wing has so much sway that it can start chipping away at these decisions it has long hated. It wasn't because the county as a whole supports this conclusion, or because their arguments were better, or anything like that. It's because conservatives got lucky with when SCOTUS nominations came up, and were particularly ruthless in making power plays.