So you've been stopped by a police officer, and they're currently investigating you for a DUI. You've been asked to step outside your vehicle, and you're not free to leave. They're going to be asking you questions, and most likely will ask you to submit to a series of field sobriety tests. You know your rights, so you inform the officer “I won't be answering any questions until I've had a chance to talk with my lawyer.”
What happens next? Is the officer required to now go contact your lawyer, if you have one? In fact, they aren't, at least in this situation!
When you assert your “Miranda Rights,” i.e. your right to legal counsel, an officer must stop interrogating you immediately, but that doesn't mean that they have to stop investigating you, or can't arrest you then and there, if they believe that they have probable cause to do so.
Your right to counsel only applies to interrogation by police officers, or statements you make while you are under “custodial interrogation,” which is a complex legal standard in its own right.
If an officer continued asking you questions after asserting your Miranda rights, or interrogates you while you're in police custody without providing you with your Miranda Warnings first, there's a very good chance that anything you said wouldn't be admissible in a trial.
This is the main point of this constitutional right: if you are in custody and being interrogated by officers, any statements or confessions you make cannot be used against you unless 1) you were informed of your right to counsel and have not asserted it or 2) you have asserted it and your attorney is not present.