Here is the methodology :
Go to the layers property menu for your contour Shapefile in ArcMAP. For the test string label select your zvalue attribute (z value is the cartesian coordinate for elevation/altitude). This layer may not be named in the same way as you see in the image below. But it will most likely carry a name such as Z value, elev, elevation etc... This is the attribute you want the line to show . eg. 250 meters

Select the following setup: Orientation curved; Postion Above(check), On the Line (check)