Technical Condenser tech

Back in the old days when we did a tune up we checked the condensor with a micro-fared tester. It either failed or passed. I am such a pack rat I still have one LOL. Mike

 

Let's break it down a bit.
When the points open, current flow stops.
When the current flow through a coil stops, a high voltage is induced in that coil, basically trying to force current to keep flowing.
If you have no capacitor the (relatively low voltage) beginning of this voltage spike will be enough to cause an arc at the points, letting some current keep flowing through the coil. The voltage spike never gets very high, just enough to ignite the arc, and the magnetic energy in the coil is used up just to keep current flowing.
Since the voltage spike in the primary winding remains low, the voltage induced in the secondary winding also becomes low = not enough for a proper spark on the plugs.

Now, with a capacitor:
When the points open, current flow stops.
When the current flow through a coil stops, a high voltage is induced in that coil, basically trying to force current to keep flowing.
With a capacitor, the beginning of the voltage spike is dampened, the voltage rises slower as it takes a moment to charge the capacitor. Not much slower, but enough to let the points open enough to prevent an arc from forming.
Without the arc the current is basically stopped completely (except for a very small current charging the capacitor, it is basically irrelevant at this point), and that means the magnetic field breaks down quicker, and induces a higher voltage in the primary winding.
The several hundred volt spike in the primary winding induces a ~100 times higher voltage in the secondary winding, and a spark forms at the spark plug.