Why is it better to castle than leaving the King where he was?

First, notice because the center is heavily contested*, it's harder to move through.

So castling is safer because it's easier for you to bring defenders from your queenside (in the case of 0-0) than it is for your opponent to bring attackers from his queenside. Your Q-side defenders travel your uncontested first 3 ranks, while their Q-side attackers must travel though or around the center.

* The center is contested because central control affords the ability to post pieces in the center where they are free to quickly maneuver to any area of the board.

Defendin' th' king.

you basicly have to casstle in like 95% of the games you play. sometimes whne you play an opening like the hippo you dont have to casstle but it is still not bad.

You may want to delay castling if it is undetermined whether your opponent is committed to short or long side castling.

It's better to castle for a number of reasons. For one thing, the King is a huge liability in the center. The number of forks, skewers, sacs, and pins you have to factor in make calculations a real headache, ESPECIALLY when the game opens up. It's just so many possibilities to keep track of. Once you get to the level where you're thinking about your moves and looking ahead you will see what a pain it is to always be forced to take your king position into account before you do anything - not to mention factoring in future safety once you castle, so it's like you have to take your present and future king position into account! You have to think about these things with every move you make. So if I haven't castled by move ten my brain is usually splitting, and because I'm a weaker player, I miscalculate and end up getting royally screwed (pun intended). So I castle as long as I'm not moving into the line of fire.

If you don't castle, you cannot connect your rooks, which is pretty important to back rank defense. 

then again. The king is an attacking piece. How is it ever going to see play when you tuck him away in a dark corner? ;)

You don't have to castle to do this, you can lift the K.

I can describe that game of yours perfectly:

https://www.youtube.com/watch?v=qVW3R5ehrvo

Here is my worst game. It came when I just started playing online. This was one of my first blitz games.

http://www.chess.com/livechess/game?id=1078782309