Here's my simplified example: Let's say I have a car, it's worth $40,000. I agree to lend it to you to drive for 3 years for $20,000 (These are the terms). You agree. So now of that $20,000, do you want to split that up over 3 years ($555.55/month) OR... Actually I don't really trust you can pay me $555.55 a month. I need you to pay me $10,000 up front and the the remaining $10,000, you can pay over the 3 years (277.77/month). You agree to this.
So now you drive off and in 3 months you total the car. Since I own the car, I need to get my money back. Your insurance company will pay me what they think the car is worth. They think my car is only worth $35,000. I don't really care as long as I get my total value back. I already have $10,000 from you, the insurance is giving me only $35,000. So I will deduct the left over amount from your $10,000. Now I will give you back $5000 only, not your $10,000. SO you lost $5000 just from having the car for 3 months. So it's important that you try to get the insurance company to value it correctly.
So now that everything is taken care of, the lease is terminated and finished. I am willing to lend you another car at a new term...
Now... If you had the option of no down, you would have been paying $555.55/month. So in 3 months, you paid me $1666.65. You totaled the car. Your insurance is only paying me $35,000. So you paid me $1,666.65 already (technically less because principle and interest, etc but just keep it simple). I'm owed $3,334. You'll have to pay me UNLESS you have GAP insurance. the GAP insurance will pay me this remaining amount. So in this case you are only out $1666.65 for the 3 months you leased the car.
Obviously this is very simplified but idea is the same.