Free radicals are extremely reactive and can form bonds in undesirable ways in cellular chemistry, which is generally bad for the organism, potentially causing all sorts of cell damage.

Cells produce anti-oxidant defenses to regulate free radicals such as the delightfully named "Superoxide Dismutase" enzyme:

https://en.wikipedia.org/wiki/Superoxide_dismutase

As an aside, there is poor evidence that consumption or topical application of anything artificially high in anti-oxidants does any good for healthy people, although you see "antioxidants" used to sell crap all the time.

Maintaining the right balance between free radicals and anti-oxidants is the sort of thing biology is pretty good at on its own. Trying to put a finger on the scales in one direction or another is probably pointless at best.

When thinking about it it seems pretty weird. Wouldn't most free radicals either enter via lungs or digestive tracts? And those will have plenty of stuff to react before any anti-oxidant can get to work... And anti-oxidants spreading through body on other hand seems bit worrying.

Most free radicals are products of reactions inside the body and the body does have mechanisms to deal with them, even if not 100% sometimes.

My understanding is free radicals are literally atom sized bombs. Bombs are trouble because they lead to dangerously quick oxidation, transforming a solid into a gas in milliseconds.

Now consider you are a cell. A lot of things you consider important are individual molecules. Free radicals are oxygen ions. So get a few of those ions inside of you and your important molecules are getting oxidized. They’re getting ripped apart. The same as a grenade in a trench. But at the indivisible scale.

Free radicals - in this case often oxyen or hydrogen on its own - are highly reactive as others have said. What that means is that they'll grab atoms from anything ripping pretty much any other molecules in the area apart, and releasing a punch of energy when they do that, which could do further damage.

Free radicals will cause cell damage and thus death or cancer