what is a finite subcover?

I'm trying to write up a proof and getting really stuck on this idea of "finite subcover."

The problem is to prove that given any closed and bounded interval [a,b] in R, any cover of [a,b] with open intervals contains a finite subcover.

I saw a proof by contradiction sketched out that went something like this:

Assume that there exists a cover of [a,b] that does not contain a finite subcover.

Divide the interval [a,b] in half. We know that one of those subintervals contains a cover that does not have a finite subcover. Divide in half again.... repeat process until we have a series of nested intervals all with the property that they are assumed to have an open cover that does not have a finite subcover.

The nested interval principle tells us that that the union of infinitly many nested subintervals will contain a single point. And here there is a contradiction that rests on an understanding of finite subcover. And that is where I'm stuck.

Can anyone please explain what a finite subcover is and why there is a contradition here? Is it that any open cover of a single point must have a finite subcover? Is that all it is?