It's a different skillset, although knowing programming (especially website programming so that's mostly JavaScript for frontend, while backend is commonly JavaScript or Python) can be helpful, having some idea of the type of errors people usually make when programming is the thing that's helpful.

This short version for how what most people call hack websites is that they figure out a security bug somewhere on that server and exploit it to gain access to that computer, and then they can take down the site or modify the contents. Doing this is very specific to the site in question and the vulnerabilities it might have, asking how people do this is kinda like asking how people break into houses, there are a million and one ways to do it and it depends on both the attacker and the target.

However a very common tactic is to point a bunch of computers towards that site so that the amount of traffic is high enough that legitimate users can't get through, that's not hacking the website though, it's the equivalent of calling a phone number nonstop so no one else can reach it, this is called a Denial Of Service (DoS) attack, and it's usually Distributed across several attacking machines as nowadays if a single IP hits you too hard you block it and carry on serving users, which is why this is known as a Distributed DoS or DDoS attack.

Finally, how can people claim to do that with impunity? The short version is they can't. These people usually take pseudonyms so as to not be traceable, and claim authorship under those, not their real names.