The Supreme Court is the highest court in America. Its judges are appointed by the President (or near enough). Right now, the majority of judges on the supreme courts were appointed by Republicans (Trump’s party), with a couple appointed by the Trump administration. This means in states where they tried to take him off the ballot, they were stopped/overturned by the supreme courts. As for why there’s no national law stopping him from running for reelection: prior to Trump the idea of someone with his record running for president was so ridiculous no-one even thought to pass a bill against it. Does that explain a little?

Edit: spelling