Not all of it interacts with us
Electromagnetic radiation comes in lots of forms, from gamma rays with short wavelengths to radio waves with long wavelengths. Some we are aware of, such as light in the visible spectrum, or heat from the sun in the infrared part of the spectrum. These types of radiation are interacting or reacting with compounds in our bodies, causing signals to be sent to our brains which then register light or heat.
Other types of electromagnetic radiation either don't interact with our bodies (e.g., radio waves) or else damage our bodies but not in a way that we are necessarily aware of at the time (e.g., gamma rays).