Let’s say you measure distance in kilometres. You’re describing a rectangle to me and you say “this rectangle has sides that are 1 and 10 units long”. This statement means very little to me if I don’t know what units you’re using.
Now instead say you tell me this rectangle has an aspect ratio of 1/10. This is a dimensionless property that doesn’t depend on the units you’re using, since they’ll ‘cancel each other out’. If I measure in metres or feet or nautical miles then I’d arrive at the same number.
There’s a sense in which this dimensionless number captures something about the essence of the rectangle. An aspect ratio of 1 is always a square, for instance.
This concept can be extended to capture more complex properties of physical systems, and making them dimensionless means you can calculate a single number and apply all of your usual intuitions to the problem, without having to ever worry about units or scale.