No person shall be held to answer for a capital, or otherwise infamous crime, unless on a presentment or indictment of a Grand Jury, except in cases arising in the land or naval forces, or in the Militia, when in actual service in time of War or public danger; nor shall any person be subject for the same offence to be twice put in jeopardy of life or limb; nor shall be compelled in any criminal case to be a witness against himself, nor be deprived of life, liberty, or property, without due process of law; nor shall private property be taken for public use, without just compensation.
When you are told you have the right to remain silent, people are referring to the 5th Amendment of the Constitution. You have the Right to Remain Silent, means no one can force you to talk to the police.
They can trick you:
“If you just tell us what happened, you can go home“
They can sound sympathetic:
“We just want to know what happened, and we can’t help you, unless you tell us the truth“
They can lie to you:
“We have three witnesses that identified you“
But they cannot force you. This means it is up to you to just remain silent.