It is a measured distance along a route. A mile marker along a highway that is nominally 10.1 miles from some start (such as where the route crosses into a county) may not be exactly 10.1 miles. Someone surveys where certain landmarks, mile markers, or buildings are along the route exactly. These route events get an M value representing their measured location. A route feature has both its line and its measured locations, and other events, along it. A measured distance along a line has many uses, such as route planning.
I need to vamoose for the day and I can't get the ESRI link to load up. I may be able to explain this better tomorrow.
(Added). Measured distance points can also be used to improve linear referencing. Some points that have been located exactly can be given higher weight, so that they have more influence on where new geocoded points are placed along a line. This is very important on long lonesome stretches of highway where there is not much to geocode from.