In mathematics, the "radius" of a circle is the distance from the middle of the circle to the outside:
In the example above, the newscaster says that residents "within a 5-kilometer radius" had to leave. That means that anyone who lives 5 kilometers or less from the volcano has to leave.
People talk about things that are "within the radius of" something important. For example, you can describe the things that are close to where you live or work:
I love living downtown. I've got everything I could possibly need within probably a two-mile radius.
You'll also hear "within a ___ radius" when people are talking about bombs or other dangerous things:
It has a blast that will flatten anything within a hundred-foot radius.