Can Someone Explain the LP system to me?

Generally that's the case. But if your MMR is out of wack, and you have say, Silver 4 MMR and you're in Gold 3, The system is trying to bring you down to Silver 4 where you belong, thus gating your LP heavily. Now, if you have Gold 3 MMR and are in Silver 4, Your LP gains will be huge, and you'll likely skip divisions as you climb.

If you have Gold 2 MMR and are in gold 3, you'll have net positive LP gains.

The system is trying to place you in your proper division. There are lots of sites that pull ranked data and do a rough estimate of where your MMR is, and thus that allows you to kind of guess where you should be placed, and how much LP you should be earning.

No. If you're doing poorly, you should be a lower rank. If you're doing well, you should get a higher rank.

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MMM okay.

Wouldn't it be a smarter system to do the opposite? If you're able to prove yourself versus "better" players, shouldn't be more deserving of LP?

I think you may be misunderstanding it a bit.

Say you have the MMR of Gold 3, but you are currently ranked in Silver 3. Your MMR is ensuring you continuously play against people within Gold 3 and your win rate has been at a consistent 50%. This clearly means you belong in Gold 3 - even though you're ranking in Silver 3 (since you win half of the time and lose half of the time - that's fair). In this case, you would earn more LP for a win than you would lose in a loss. That way, you will eventually hit the ranking you belong in - Gold 3, as this is where you are consistently performing.

However, your case appears to be the opposite. Your ranking is higher than the skill level (MMR) you are being matched at - so the game is trying to bring you back in line. The system is trying to align your ranking with your MMR. So if you continue to win and lose at 50%, your LP gains and losses will eventually equal out (assuming you do not become a more skillful player).