Honestly, when first starting out, I was very confused. There is a lot more to chess than I thought, and that chess takes practice. I feel like it is an experience game and not an intelligence game. 

Correct!

The two biggest misconceptions about chess are:

1) The winner is more intelligent or that to be a very good player requires high intelligence.
2) The winner is the person who calculated more (or saw further, planned farther, etc).

Chess is a skill like anything else. Think of painting, fixing cars, speaking a new language, sports, a musical instrument. Highly skilled people are those who put in many hours of work. They might also be smart, but mostly they put in tons of hours over many years.

As for why chess is so hard, here's something I like to say.

Chess isn't hard because good moves are hard to find. In most positions when you're shown the best move it certainly looks good.

Chess is hard because lots of bad moves look good too, and we become distracted by them, and play them sometimes.

So, especially when starting out, one of the best things you can learn to do is challenge your candidate move. Imagine it as if it's been played, and try to find everything wrong with it (because that's exactly what your opponent is going to do after you play it). Specifically look for checks, captures, and threats (this is Dan Heisman's advice). If you still like your move after finding the worst things about it, then you can play it.

This basic idea for newer players is probably the best analogy to every day life, and the only real link to something people consider intelligent. In real life if you like an idea, it's not enough to find reasons to continue liking it, you have to actively seek out the strongest criticisms against the idea you like, and decide whether you have a good answer for them.

That gets you past beginner level... everything else is chess specific and doesn't really have any real world application

Because you are exposed to so many possibilities when you started learning it, unlike schools where students typically embark on a primary or lower level education at first (with easier-to-understand concepts) before proceeding to more advanced ones.

Easy to play, difficult to master someone before me aptly quipped!

Chess is easy to play. Winning is the hard part. Many players are more dedicated than us to study winning Tactics and Strategies.

What's the Russian saying?  Something about chess being a lake from which a gnat may drink and an elephant bathe...  

Indian saying, but yeah.

"One doesn't have to play well, it's enough to play better than your opponent" - Siegbert Tarrasch