The earth spins as it's rotating around the sun. That means if we drew a line from the sun through the center of the earth, that line would hit a different part of the planet every moment that passes.

It just so happens we've divided the day into 24 "hours", and the sun hits the same place twice if you wait that long. (Technically it's a tiny bit different from 24 hours but that's something that only matters once we get into higher-end calculations.)

So the "time zones" are made by borders that are ROUGHLY where that "direct" line from the sun hits an hour apart. I say "ROUGHLY" because these are also political boundaries so they're set by people who vote or own land etc.

This was done for our own timekeeping convenience, so that we could understand questions like, "When is it morning on the west coast?" easier. Imagine if we made everything use East Coast time. That would mean we'd have to remember that "morning" in California will be somewhere around "noon" in this time.

Is it really easier to remember, "California is 4 hours later than NY so at 8AM in NYC it is 4AM in California"? Well, that's subjective. But this is how we did it, and it'd be at least as complicated to do it the other way.

The main place where this makes the most sense is day boundaries. We understand when the time changes from 11:59 PM to 12:00 AM the date has changed. But if we didn't have time zones, some people's day would start at 1AM. Some would start at 2 AM. Everyone's day would roll over at a different time of day. They'd have to show up to work at a different hour, etc.

From that perspective, even if you travel to a far-away place, it's easier to remember, "I need to be at the office when all the clocks around me say 8AM" than it is to figure out, 'OK, here the workday starts at 7PM so that's when I need to be at the office.'

So, put short: it ties our time to roughly when we expect the sun to rise and set. If we didn't have time zones, in some places the sun would rise at midnight, and the name "midnight" wouldn't make a lot of sense if associated with 12 AM.

Edit

To answer your question: yes, they record multiple intros. Most of these shows are mixtures of live TV and pre-recorded cuts, and often they cut to other actors. So they might do the startup of the show with a pre-recorded introduction, then the title jingle, then move to live footage. Or, in the Eastern time zone, they might be running a commercial break while the host does a live introduction for the Western time zone. Then when that introduction is finished, the Western time zone gets commercials or another segment, and eventually everything syncs back up to live footage again. Sometimes "live" shows like this aren't actually "live" in all time zones because of these logistics.