One of the most damaging characteristics of free radicals is the domino effect, or cascade, of damage they cause. A free radical is an unstable atom or molecule. Usually oxygen based. An unstable chemical interacts with the first thing it touches in an effort to acheive stability.
This interaction causes damage to the substance (a part of the cell) it used to become stable. The damaged substance is now also unstable and it needs to interact with another substance. The substance it interacts with is typically an adjacent part of the cell. Then that substance becomes unstable and on it goes.
When a cell structure becomes unstable, it is damaged. When it interacts with an adjacent substance to become stable again, the damage remains. So, like a domino effect, a free radical damages one part of a cell and sets off a chain reaction of destruction that decreases function and ultimately leads to cell death.
All of the instability and chain reactions described here are on a chemical level. An atom or molecule is unstable if it has an unbalance of protons and electrons.