I like the first answer very much but I thought I'd provide my own take on the question with some of the cases in the show I could remember.
The best example off the top of my head is the episode Murder Under the Glass where the killer, who is also a renowned cook, poisons his victim when he dines with him and Columbo, after busting him at the end, tells to him that he figured out he is the killer as soon as he met him because he was surprised to see him arrive at the death scene immediately and not to see him go to the doctor after the person with whom he had dined died of food poisoning.
In another episode, Death Lends a Hand from Season 1, when Columbo investigates the body of the victim he finds the cut on her lip from a slap (and also the first person on the police team to notice that) and in the first scene where he is introduced to the killer by the victim's husband he says to him he can read palms and by that I think was able to make the deduction by looking at his big ring on his hand as soon as he meets him.
There were other instances like the episode Negative Reaction where the killer kills his wife and tries to frame another person by killing him in a ransom exchange which he schemed, but Columbo gets the facts from the scene of murder where he investigates a witness who claims to have heard the shots when he stated that there was a considerable time lapse between the shots and when he asks about it to the killer his justification pretty much isolates him as the prime suspect.
Also the case with The Art Collector where he was so focused on setting up his alibi at an art show after he kills his uncle by asking time from almost every key person in the art gallery which immediately puts Columbo right onto him immediately.
If you look at all the Villains in Columbo they are mostly successful people on their profession, celebrities and most importantly incredibly intelligent that they cover up all the major loose ends and their alibis but they get caught on much simpler detail they might have overlooked which was exploited by Columbo and his dogged persistence.
The best quote I could think of coming from Columbo about himself its from Bye Bye Sky High IQ case where he says to the killer:
All my life I kept running into smart people. I don't just mean smart like you and the people in this house. You know what I mean... I could tell right away that it wasn't gonna be easy making detective as long as they were around by working harder than any of them, reading all of the required books and paying attention to every detail.