The incenter is where the angle bisectors meet. This is a point of concurrency that must lie inside the triangle. If you picture what an angle bisector looks like and how to do a rough freehand of one, you're basically keeping your pencil squarely between the sides of a the angle that you're bisecting. Hopefully it makes some intuitive sense that this is running smack in the middle of the sides of an angle. If we extend the idea a bit and do that with all the angles in a triangle, we'll end up in a place that is equidistant from all 3 sides of the triangle. This is also the same as the center of a circle that is tangent to (or touches) the sides of the triangle.
