I think there's two types of answer to this question. One we see on screen and one we can only speculate about.
I've often wondered how many of Columbo's cases would have collapsed in court.
In the case of Barry Mayfield for example, what would Mayfield's defence have made of the incriminating evidence turning up in Columbo's pocket?
Columbo was always good at finding means, motive and opportunity but in some cases you never actually see him prove that person carried out the murder.
Even though I'm one of its defenders, there's MIND OVER MAYHEM.
In spite of all the blunders Marshall Cahill makes, what makes him confess is Columbo arresting his son on a trumped-up charge (which he later ADMITS is a trumped-up charge), so why WOULD'T Marshall confess? (A little like Ned Diamond confessing to cover for Grace Wheeler, except that it's covering for someone who's actually innocent.) And according to Columbo, he probably couldn't get Marshall any other way, as in evidence.