What does the sentence of "let's get the party started" mean? Or in general what does the meaning of get sth past participle?
In "let's get the party started," the word "started" is used as an adjective. (It's a participle.) In "let's start the party," the word "start" is used as a verb.
In effect, since "'starting the party' gets the party started," the two are interchangeable. In some situations, phrasing the sentence so that "started" is the last word is preferable as it strongly emphasizes "started." On the other hand, the way "gets" is used in that form is not very formal.