Chapotraphouse

They are part of normal police. It stands for Special Weapons And Tactics. They are more heavily armed and armored compared to regular cops. They have armored vehicles, better training, and more stringent fitness requirements. The only time I saw a SWAT raid up close they closed down my street and had two snipers because a man robbed a nearby gas station convenience store with a shotgun. They killed him.

Here's a 2015 quote from economist.com. Budgets expanded beyond belief from $1m in 1990 to nearly $450m in 2013. And the number of times they were used exploded 17 times. 3,000 in 1980 to 50,000 times.

→ The American Civil Liberties Union found that the value of military equipment used by American police departments has risen from $1m in 1990 to nearly $450m in 2013. The equipment has been used. In 1980 SWAT teams across America were deployed around 3,000 times. Deployments are estimated to have risen nearly seventeen-fold since, to 50,000 a year.

Of course - 2015 was ten years ago so things have surely gotten even worse.

How does SWAT rapid response actually work? Are there a bunch of guys constantly ready to go in a warehouse somewhere?

depends on the department, but usually the pigs in the department's swat team(s) are also regular cops and they get notified by dispatch when they are needed if they are clocked in, or literally paged if not
then they put on lights and sirens to get to the station quickly and arm up

What even is SWAT, are they part of the normal police?

heavily armed response teams and yes, usually