He confessed.
As with many episodes, there is no "arrest".

He told Columbo that he did it and he'd do it again, so yeah, he's going to jail. There are a lot of cases that will get a conviction due to either the evidence or they confess, but there also a few that a good lawyer will get the murderer off.

Rumford's mistake WAS confessing. If he hadn't, it would have been a tough conviction.