They went to Washington because Trump specifically asked them to come on January 6th. It was not a spontaneous event.
I agree that is the reason for the WHEN and WHERE. The reason WHY a person shows up varies from individual to individual. The man with a truck full of Pipe Bobs two blocks away came to DC for a different reason than the man with the Zip Ties. Both of them came for a different reason than the man who was just talking to his wife on the phone when he had a heart attack, or the man that liked to sell stuffed Kangaroos that were dressed like Donald Trump. Some came to support a president they loved. Some came to protest an election they believed involved cheating. Some came to kill and destroy.
President Trump wanted them there to put pressure on Congress.
Almost certainly. However President Trump also wanted all 300 million Americans to vote for him, so what "the Donald wanted" is irrelevant. He invited many, and those that came did so for their own personal reasons.
He did it because his false claims of voter fraud could not be supported in the courts (five dozen lawsuits).
This is one of those dangerous HALF-TRUTHS. I will completely admit that his legal war to overturn the vote was a complete failure.
That said, MOST of the cases were not decided against TRUMP, but dismissed without being heard for procedural reasons. That means that Fraud may or may not have happened, but the person bringing the case before the court is not the person to be bringing this case at all (or this is the wrong court to hear that case). That counts as neither a victory nor a defeat on the strength of the facts. [Those are the cases that bother me, because I want Biden proven innocent and the election proven fair ... not "we don't want to look".]
Some of the cases were utterly shredded and shot full of holes. The accusation of NO republican watchers present in a counting center and Republicans attempting to enter was embarrassingly debunked as completely untrue when it came time for sworn testimony. So there was NEVER a time when any Democratic "machine" could have done whatever they pleased completely unobserved. That was good to hear because that is EXACTLY how the system was supposed to work.
Some of the cases ruled "yes, they cheated like a Vegas card shark" but we have no evidence that it happened EVERYWHERE and would effect the election. The cheating was not widespread.
Frankly, it is not the JOB of private citizens to investigate reports of organized crime and election tampering. We have an FBI and Justice Department to investigate serious crime. There are enough serious statistical anomalies that they SHOULD be investigated so the rumors can be refuted with better facts than "you can't prove it happened".
["You can't prove I smuggled cocaine over state lines, and even if you could the statute of limitations has passed" is not the same as "It never happened".]
They believed many lies because there was no way to discern which statements were lies and which were truth. Both sides were (and are) lying, which fans the flames of misinformation.