Leverage is an investment term that means using borrowed money to increase the potential return of an investment.
When Buffett says: "Thus, a $20 trillion asset class, housing, out of the nation’s $50 trillion in assets became increasingly levered up." he is saying that this asset class (housing) is increasingly relying on borrowed money. That is more and more these assets (houses) are being bought with borrowed money.
He is implying there is a risk here. Investors borrow money to buy their houses assuming their house will go up in value more than the cost of their borrowing. But if say interest rates on borrowings go up and house prices go down or stagnate, then the borrower is at risk of being forced to sell his or her house and house prices could tumble, so the $20 trillion house asset class could burst like a bubble and be worth much less.