The work of stepping up is done in the ignition coil, which it not just a coil, but actually a transformer, which has a high turns ratio.
But we know that you can't directly transform DC through a transformer. So we must have a way of making a 'psuedo-AC'. How it works is that the 12VDC is allowed to flow through the primary, which sets up a magnetic field in the coil. Then it is quickly interrupted (either by mechanical switching, or electronically). The magnetic field collapses rapidly, and as it does so, it cuts the secondary coil. From basic electrical theory, we know that when you cut a coil with a magnetic field, it induces a voltage in the coil. More turns = higher voltage. Hey presto! You got 10-20kV! 8) This process repeats itself for each power stroke.
these "mini" transformers are great for contructing "jacobs ladders. Be warned though. These arcs ionise the air and forms nitrogen dioxide (NO2), which is poisonous.