Radon is a "daughter" or progeny of the radioactive decay of common radioisotopes in the environment. Uranium-238, a fairly abundant radioisotope in the Earth's crust (it has an extremely long half-life---about 4.5 billion years), decays to radium-226. Both these materials are solids, so they are immobile in soil. Radium-226 has a half-life of about 1600 years; it decays to radon-222, a gas.

Radon gas can now migrate through the soil. Normally, it would just seep out into the atmosphere, where its ultimate concentration would be so small, it is almost impossible to measure. However, the improved sealing of an energy-efficient home causes the air pressure indoors to go up high when the air warms, and decrease when it cools (there is a "heating/cooling cycle" going on inside the house from the thermostat turning a furnace on and off, which causes the air temperature inside to cycle up and down). During the times of lowered pressure, air is sucked inside from the soil through cracks in the basement walls or around pipes that have to enter the house. Radon gas is carried in with this air.

Once in the house, the radon decays to polonium-218 and then to radioactive lead (radon has a half-life of about 4 days; polonium and lead-214 have half lives of minutes). Polonium-218 and lead-214 are solids, so they exist as aerosol particles.

Given the difference in the half-lives, it is more likely that polonium-218 or lead-214 will radioactively decay while inside your lungs than radon. It is the ionizing radiation from radioactive decay events that create the increased risk of developing lung cancer. Thus, the progeny of radon are the likely culprits, not the radon. The radon just provides a means for radioisotopes to migrate from the soil into the air inside the house.

One other entry point is through the water supply. Radon is water-soluble, so groundwater probably has some radon gas dissolved in it. If that happens to be your water supply, then when the water runs out of a faucet, the aeration/agitation will help degas the radon from the water. You can also drink the water and ingest the radon atoms.