I always hear that he is either the best or or he the worst. What is the basis behind these claims?
Let me make a quick argument in favor of FDR being one of the best.
FDR enters office during the worst economic crisis in American history. Markets are tumbling. People are rushing to pull their savings out of their local banks. He responds to this by declaring a bank holiday - which slows the bleeding slightly. Then, over his first 100 days in office, he pushes forward a series of new laws - many of which are still with us today. For example, our savings accounts at the bank are insured by the federal government up to a certain amount because of these regulations passed by FDR right away during his first term.
Ok, so the guy makes mistakes along the way. He isn't perfect. He tries to unconstitutionally expand the Supreme Court to tip the balance away from more conservative judges. Some people have argued his early reforms were less of a coherent set of ideas than throwing things at the wall to see what sticks. In my view, given the desperation people felt at the time, this isn't a devastating criticism. Some of the ideas failed while others had a big impact.
Over time, he creates what he calls a "New Deal" for the American people. This results in dozens of new programs and agencies (sometimes referred to as "Alphabet Soup" agencies for their propensity toward acronyms. WPA, CCC, ERA, etc. Historians debate the exact effect this had on the economy, but it's almost beside the point. Having jobs had a major impact on morale during the depths of the Great Depression. It also resulted in the largest overhaul of our infrastructure (roads, bridges, schools, post offices) that we've witnessed in American history before or since. FDR also took a page from his relative Teddy Roosevelt in greatly expanding the National Park Service - an under recognized and major contribution.
Check out this amazing map visualizing the New Deal, it's actually pretty staggering - https://livingnewdeal.org/map/
In one oral history interview, an African American man remembered, "Franklin D. Roosevelt saved this country from a [violent] revolution. If he hadn’t done what he did, look where we would have been. We would have riots. I mean real ones. Shooting. But he held it until Hitler come along. Then they opened it up."
As the above quote suggests, FDR then guides the country with a steady hand through the largest conflict in human history (WWII).
He was extraordinarily politically savvy and could convince politicians to support his efforts when he needed them. He was widely known to tell contradict himself by telling two political opponents they would get they wanted (often leaving one party surprised and pissed off by a decision). He was immensely popular with many people for providing jobs and food during a time of deep crisis and connected deeply with people through "Fireside Chat" radio addresses. He gave unprecedented access to the press in exchange for downplaying his physical disability resulting from polio. These things fed into his popularity.
FDR was heavily criticized for breaking with Washington's precedent and not stepping down after his second term in office. Conservatives hated that decision. He also probably carried on sordid love affairs.
Ultimately, however, most historians agree that given the circumstances and conditions he encountered in office, FDR proved to be one of our best presidents ever.