It goes deeper than anecdotal "I wish my client hadn't said anything", there are concrete reasons for this advice.

For example, according to rules of evidence, any exculpatory statements you make to police are hearsay and therefore cannot be mentioned in a trial. The police _can_, however, try to use that information against you and then use information derived from your original statement at trial.

There's a pretty popular lecture[0] on this. It covers a lot of edge cases and common arguments against "don't talk to the police".

[0]: https://www.youtube.com/watch?v=d-7o9xYp7eE

My comment was based on what I've heard law enforcement say on the matter. If someone is in the position where they've been given a Miranda warning, they've been detained and they know they're in trouble. People tend to think they can talk themselves out of trouble, which is rarely the case. Of course, I've also heard that plenty of suspects will start saying incriminating unprompted when you throw them in the back of a police vehicle.