If you just had normal insurance, your insurance company will pay you thye market value of your current car, with that money going to cover what you were obligated to the leasing company.
General example: So if you're leasing a 2010 Whatever and you have $300/month payments and you have 1 year left on the lease, that would equal to $3600 in remaining payments, after which if the buyout was $10,000, you would owe basically $3600 plus $10,000 = $13600 to the leasing company.
If the market value of your car at the time was $12000 and your insurance company gives you that $12000, then you owe another $1600 out of pocket to the leasing company for the car. If market value for your car is $15000 then you give $13600 you owe to the leasing company and keep the remainig $1400.
This is important though: Market value, basically the amount you'll get from insurance, has nothing to do with what you owe on the car, that's why you could end up breaking even, owing out of pocket, or in a rare case getting to keep some of the insurance money if your payout was less than your market value.