There does exist train service between San Diego and Los Angeles, it's called the Amtrak San Diegan. The problem is that in Los Angeles it stops at Union Station in downtown LA, which isn't exactly close to LAX. There may be a bus between Union and LAX, but staying on rail lines you would have to take the Metro Red Line subway to another LA station, then the Metro Blue Line to the Metro Green Line, which has a bus running from one station to the terminals at LAX. But you can't have a lot of baggage on the Metro Rail.
Once you've done all that, it's not much more money to just fly (and maybe even cheaper; my next tickets to fly SAN-LAX are only $56 round trip on United Express).
As DC-10 pointed out, most people doing this aren't just flying from SAN to LAX, they're going on somewhere else. I'm one of those people.