Not exactly milli-radians Russ. close though.
Nato Mils are layed out as 6,400 mils in a circle. I think the Swedes used 6,300 and the Russians used 6,000.
If it was actually milli-radians it would be 6,283

It helps to think of them more like %grade.
Surveyors often measure grade as a percentage instead of in degrees. 1% grade is one foot change in height over 100 feet of distance.
With mils at 1 kilometer distance 1 mil is 1 meter, or at 1,000 feet it is 1 foot.
Since you are using it on a compass mils can be read like degrees and you can relate them directly to a measured distance.
(It is accurate enough at narrow angles but it distorts as the angle gets wider. The problem is radians are measured along the arc of the circle but straight line measurements are tangents or chords of the circle. For large angles you need to correct using trig tables.)

This make it possible to do range corrections like JB says and you can also use it to estimate ranges to targets.

To explain that:
If you know that a man is really about 2 meters tall and he measures 2 mils tall in a telescopic sight then he is about 1,000 meters away.
You can use other objects like buildings or vehicles the same way.
If you look at a tall building you can assume 3.1 meters floor to floor for residential and 3.9 for office buildings.

Of course with a compass you have a hard time measuring heights but there are similar rules on horizontal measurements.
For example most transport truck trailers are 53 feet (16 meters) long.
If you have a few other common object sizes to use you can estimate ranges fairly accurately.

Remember that it isn't just for shooting but for any measurement at a distance. You can even use topo features instead of man made features if you have an accurate map.

I am not a user of rad, grad or mil measurements because I am usually working in degrees with lat and long.
However on a map with Easting and Northing (The Civilian UTM Grid Reference System or the Military Grid Reference System)
the grid on the map gives you the location as a distance measurement from the west going east and from the south going north.