I'm an engineer, so...
Short answer: It depends.
Well, there are a couple of really important things missing in the above answers.
-> How can a capacitor work in place of a car battery?
The first thing that I notice here is that everyone is assuming that the capacitors are only being used to start the car. A car battery serves 3 main purposes. Starting the car (high current, short duration), running equipment within the car while it's running (generally low current, but also very low efficiency required because the alternator takes over for most of it, and uses the battery mostly as an additional regulator), and, finally, keeping the "passive" (as in, not required to run the car) systems like the clock and the lights working when the engine is off (low current, long duration).
A capacitor is fine for the first two, but really sucks for the third (and often most important) one. Ever turned off the engine and needed the lights inside or out? How about turned on the car and had all of the stored channels on your radio honked up and the clock wrong, or the other settings that you've set suddenly not work because you changed batteries? Major PIA.
-> How does a capacitor work in the first place?
A capacitor works by having a dielectric (a substance that insulates/prevents electron flow) positioned between two electrodes (wires of some sort). As the electric potential (the number of electrons "attracting" to the wanting protons on the other side) builds up, the capacitance grows to the limit of the device (its "capacity"), then it either stops working or, more likely, explodes like a lipo battery. Think of it like a jug of water at the top of a water fountain... you can put only so much in before it overflows, but it's very fast and easy to refill when it gets low or empty and it can be emptied very quickly.
And, everyone forgot to mention leakage. All energy storage devices (battery, capacitor, angry wife, whatever) leak the energy that is initially stored. Some faster leakage (angry wife), some slower leakage (battery). A lead acid battery can be expected to hold a charge WITH NO DRAW AT ALL for several weeks or months in the best cases. A capacitor won't last anywhere near that long, a few days or weeks at best. The wife just leaves, so that's almost zero, but then she's someone else's problem so no worries.
Leakage in a capacitor is the result of electrons "migrating" through the dielectric. No matter how good the dielectric is it's still not a PERFECT insulator, so it WILL leak.
The past 40-50ish years of work on super capacitors is around making this dielectric almost perfect and figuring that we can work on surface area later. But, just like how a straight piece of wire still presents SOME resistance (and thus gets hot), the same is true in reverse for a capacitor. You want the dielectric to be almost zero in thickness, have maximal surface area and have an infinite resistance. But, once the distance gets down to even nanometers (not to mention, femptometers -- 1/1000th of 1/1000th of a nanometer, which is probably what's required for a true super capacitor), physics starts to go a little crazy. Quantum tunneling gets involved and things get more complicated. To give you an idea, a single hydrogen proton is about 1.7fm... and an electron is A LOT smaller than that.
So, this question and its in depth answer is a lot more complicated than it initially sounds.