Here’s the quick and dirty answer to why the rule exists: we believe in cross-examination. (I’m going to get to the operation of the rule and–just as important–the exceptions to the rule in a moment so please bear with me)
Just think about your own life experience and how many times person A told you something was true and it turns out that his whole basis for saying it is that this other guy , Person B, heard it from this other guy, Person C, and then he told Person A.
And so then you go talk to Person C and it turns out, no no no, that’s not what Person C said at all, somebody must have misunderstood, OR Person C was obviously lying, or Person C was not obviously lying, but just seems kind of dimwitted or confused.
So we have a basic rule, NO HEARSAY. If you want to get into what Person C has to say about something, then bring him into court, and let the jury take a look at him to judge his demeanor, and let both sides get to ask him questions to probe what he is saying.
Now there are a whole bunch of exceptions to this rule, each with its own rationale. The particular exception in operation in your hypothetical is the “party opponent” exception. Basically, this states that the hearsay rule does not apply when we are talking about statements made by a party to the legal proceeding. This is because a party DOES have the opportunity to cross examine the witness: he can take the stand.
Take two cases. Case 1: the out of court statement is made by the afforementioned person C. Case 2: the out of court statement is made by a party. In Case 1, the party objects and says, hey, you can’t introduce this statement by Person C, because I don’t have the opportunity to cross examine him. In case 2, the party objects and says, hey you can’t introduce a statement that I made, because I don’t have the opportunity to cross examine myself. Oops, wait a minute, yes you DO have the opportunity to cross examine yourself because you can take the stand and give your version of the story.
Now, why can’t you introduce statements that you made before? Why does the hearsay rule prevent you from doing that. Again, because if you want to say something, you can take the witness stand and say it, and then subject yourself to cross examination by the other side.