Chemically speaking, oxygen is pretty darn reactive. This reactivity makes oxygen highly useful for the chemical reactions needed for respiration, but also means oxygen can easily form free radicals.
What are free radicals? In chemistry, we define a free radical as any species with an unpaired electron. And, if you’ve ever studied chemistry, you’ll know: electrons really like to pair up.
So that molecule with an unpaired electron (the free radical) really wants to grab an extra electron, and it’ll go to any lengths to get there, reacting with anything in sight, including your DNA (which can lead to mutations and, unfortunately, even cancer).