Basically, exclamatory sentences are sentences that show strong emotion.
I love that kitten!
The sentence above is an example of a sentence showing happiness, but you can also use exclamation points to show anger, frustration, confusion, love, etc.
You name the emotion, and the exclamation point can punctuate it!
One word to the wise: Don't use exclamation points too much. You are only supposed to pepper them into your writing if at all.
If you use them too much, they will either lose their effectiveness or your audience will think that you are continually yelling at them.
Neither one of those is a good thing.
* However, I must admit that I am a rabid rule breaker when it comes to this one. I find that in my editing process, I am deleting many, many exclamation points. I guess that I use them to sound chipper, but I shouldn't use them as much as I do.
The other job of the exclamation point is to punctuate interjections.
If you don't yet know about interjections, you're in for a treat.
They are one of the eight parts of speech, and they are words that we use to show emotion. They are not grammatically connected to the rest of the sentence.
Here are some examples.
Hurray! My teacher gave me an A!
Yuck! This hamburger is disgusting!
Note that you can also punctuate interjections with a comma if they are showing a feeling that is not as strong.