Or the cops can tell you that they already have evidence which pins you to the crime, and if you confess now you can get a lesser sentence. So you freak out, not wanting to go to jail for 10 years, and you sign a confession to ensure you only get 2 years. You are still innocent, but with the threat of having your life taken away, you will do anything in desperation.
Or they claim that they have a second suspect in the next room who knows you and knows you were at the crime, and they tell you that the first one to confess gets the lesser jail time.
The reason why these claims can coerce false confessions is that, even though you are innocent, you know it's entirely possible that the court system can **** you over if you can't prove reasonable doubt. It happens all the time, especially if you can't afford a good lawyer.
The point is that innocence should have the most protection. If the cops have such conclusive proof that you are guilty, then they shouldn't feel threatened by your rights to a lawyer. The lawyer won't matter, you're still going to go down.
The onus should never be on the suspect to ensure that their innocence gets preserved; the onus is on the POLICE to provide evidence of probable cause so that charges can be pressed. If the police can't do that, they have no business pressing an innocent person for non-existent confessions or information, and that is the precise reason why rights to a lawyer are needed.