since, apparently there is no evidence that anything really
happened, why should I believe anything criminal happened?
The evidence is in the form of witness testimony from law enforcement officers, corroborated (apparently) by audio tapes, all of which are admissible in court.
What is alleged to have happened is the solicitation of sex in a public area, which is a crime in most places that I'm aware of.
The officers felt that they had enough evidence to arrest him.