Please tell me how it works and how that differs from most people's perception.
For one thing, you cannot initiate a nuclear launch with a single person.
it requires authorization by both the president AND the secretary of defense.
Neither of them has a complete code to initiate a launch.
Additionally, there are two more staff in the launch control center that need to do the same thing. They require two people to turn the launch keys at the same time, and the LCC is inside two vault doors for land-based missiles. If one refuses, the launch won't happen at that site.
For subs, someone has to give the combination lock codes to the two designated keyholders so they can even get their keys to initiate a launch.
Assuming Trump does lose his mind and order a nuclear strike after becoming president, and the Secretary of Defense doesn't agree with him, Trump COULD fire him and appoint a new Acting Secretary and try again.
In that case, the Vice President and a majority of the heads of the Executive Departments could invoke section 4 of the Twenty-fifth amendment to the Constitution and have the President declared incapacitated.
There are many safeguards against the malicious launching of nuclear weapons.
I can see that there's a slight possibility that the President could go insane and order a launch, but there's no way the Secretary of Defense, Vice President, an the majority of the Heads of State would agree with them without a SERIOUS threat to national security.
One of the major arguments against Mutually Assured Destruction as a viable means of nuclear defense is that many launch operators would refuse to be responsible for the death and destruction of a nuclear launch, even if they were under attack.
If you've ever seen the movie War Games, they show the launch protocol as it was decades ago and how one person refusing to launch can stop a malicious launch order. It's not 100% accurate, but it's pretty close.