Would it change the meaning of a sentence if I say "enhance" instead of "increase"?

Why does the speaker mention Nicolas Cage all of a sudden? Is it some kind of a joke? Or is he referring to one of his films which is about the topic they're talking about?

Yes, it's kind of both of those things! It's a joke, and they're referencing the movie "National Treasure," where Nicolas Cage's character steals the Declaration of Independence. Although, he got the context wrong in this part and later corrects it by saying the "Declaration of Independence" rather than the "Constitution". They're definitely trying to be funny by referencing this movie here. It's mostly funny because in real life, that wouldn't have happened. Ex: Nicolas Cage stars in National Treasure, where he steals the Declaration of Independence.

What's "thrilled" mean here? Isn't it like "suspenseful?"

Not quite! Here, "thrilled" means "excited;" to be extremely happy about something. Ex: She was absolutely thrilled to go on her trip. Ex: You must be thrilled about your new job!

What is skipped in front of this sentence?

"Do you" are left out at the beginning of the sentence. The sentence should be "Do you want to watch..." During informal conversations, you may notice that words are left out of sentences. This is because the remainder of the sentence can stand alone and is perfectly clear without the help of other words. Words that are commonly left out are helping verbs, articles, possessives, and personal pronouns. Ex: (Do you) Know where it is? A: How's Annie? B:(She is) Not feeling well. A: How are you? B: (I am) Doing well. Ex: (Are) You ready?

I do understand the word "recess" refers to a break time in a school. However, can I say "recess" for other situations as well? Like a break time in the work, perhaps?

Yes, you can use "recess" for other situations! It's used commonly in court when you take a break during a case. You can use "recess" when you want to take a break during a meeting. However, I would say it's less common for individual work breaks. Ex: Good job, team! Let's take a brief recess and come back to the board room in five minutes. Ex: After a month's recess, we went back to court.

Why does she say "when will my life begin", not "when my life will begin"?

She says "when will my life begin" because she is asking herself "when is my life going to begin?" If she had said "when my life will begin", the meaning would change because she would be saying that her life is "going to begin", as though she knows when that will happen.