I would like to know why the phrase "in which" is used in the following sentence? Is the use of this correct?
"I'd like to give you a little background about some of the new and revolutionary ways
in which paleontologists are looking at these ancient creatures?"
Are you familiar with the question, "In what way?" ?
Maybe in response to "I love you!"
"You love me in what way? As a friend or more than a friend?"

And even though I'm only 23 (my mother's fault), I know a Shirley Bassey song that goes...
"Something in the way he moves..."

Usually people do things IN certain a certain way.
So yes, IN is the correct preposition to collocate with WAY/MANNER.
You can use "in which," but I think "that" sounds better. It's more appropriate to use "that" to begin a noun clause.

However, you can use "in which" because it would make sense to say "paleontologists are looking at these ancient creatures IN new and revolutionary ways"