The reason for the mandate is cost sharing. Also known as the basics of insurance. When you pay for homeowners insurance and your house doesn't burn down, your premiums go to your neighbors whose house did burn down. Same idea with health insurance. Treatment for the sick is paid by the healthy. And this is the case whether you're talking about single-payer systems or private health insurance.
Let's pretend there's no mandate. You're young and healthy. Why buy health insurance? It's very unlikely that you will be hit by a bus, and you don't have any chronic conditions that require treatment. If you get unlucky and get hit by a bus, you can buy insurance then. Because there's no "pre-existing conditions" anymore, so you can't be denied.
Result? No healthy people in the insurance pool. Which means no cost sharing. So insurance gets expensive. So people in "OK" health stop buying it because it's no longer a good price for what they receive. So insurance gets even more expensive. So people in "moderate" health stop buying it. So insurance gets even more expensive. And so on until there isn't insurance anymore.
The technical term for this is "adverse selection".
The mandate keeps the healthy in the insurance pool to keep the individual costs down. Works just like in single-payer countries, except their mandate is called "Taxes".