Hmmm...assume the gasoline is entirely vaporized, it isn't, but we're dealing with ideals here. Assume all gases behave idealy, none of them do, but we're stabbing in the dark otherwise. Using n-octane:
C8H18 + 12.5 O2 --> 8 CO2 + 9 H2O
You start with 13 and a half volumes of "stuff" as a gas, you end up with 17 volumes of "stuff" as a gas -- you end up, at the end with more individual particles, which the ideal gas law says bounce around independently. This can be a little confusing, basic chemistry says that we can't create matter. But we treat each individual gas molecule as a point source of pressure, so they do take up more space.
Secondly, after combustion, you've made a bunch of hot gas products. Again, using the ideal gas law, PV=nRT, depending on the pressure, (which we don't know) the volume is the product of how much gas you have, and it's temperature, and a constant. Put more pressure on it and you can squeeze it into any space you want. But ...
OK ... summary time. Internal combustion engines work by a serious of controlled explosions. So gasoline and air mixtures can be considered explosives. Pretty much, a simple definition of explosive is: take a small amount of solid or liquid, make it into lots of gas, quickly, and maybe make it hot, so it takes up lots of space. I haven't worked out the math, as you can see, but yeah, 17x expansion of volume doesn't shock me.