The word "let's" is used when you want to give directions to a group, and you are a peer in that group. It is often used to set a plan in motion (e.g. "Let's go!") or to build momentum for a plan in motion (e.g. "LET'S ROCK!").
In this case, "Let's get the party started" is both a signal to begin the party as well as a rallying cry for the people celebrating. That phrase is also often used figuratively - for instance, before doing something dangerous, or (ironically) before starting something unpleasant.
Conversely, to "let" something happen implies you have authority. The person who is being "let" to proceed is therefore not your peer, and you would normally be using more formal language than "get this party started." If I were going to use "let" in this context, I would be more formal, saying something like "Let the games begin."
Another example of "Let" as authoritative is from the French Revolution. A famous noblewoman was told that the peasants were starving, and they had no bread. She proclaimed "Let them eat cake!". (She was later executed in the Revolution, of course.)
Contrast that with "Let's eat cake", which would have been an invitation rather than a condescending dismissal. The moral of the story is that you should learn the difference between "let" and "let's", because it could save your life!