This is an issue with using curves and a shapefile. Curve geometry is only supported in feature classes (in a geodabase). Shapefiles do not actually support curved geometry. The software still allows you to draw/edit shapefiles with the curve editing tools, making it look like it supports curves. But it will auto re-draw the curve with numerous short straight segments when you are done with the edits, thus adding numerous vertices. I think it makes these changes when you click save edits in ArcMap, but it's been a while, so I could be wrong on exactly at what point in the "done editing" process it makes that change. If you need to support true curves, you should look at moving to a geodatabase where this functionality is truly supported.
Also, for features where you already have the numerous short segments and tons of extra vertices, you should look at the Generalize geoprocessing tool and/or edit tool (see advanced editing toolbar) as it should be able to help clean that up. You may want to play around with the tolerance settings for the Generalize tool first before using it on your main data though as it can make unacceptably large changes to the data, depending on your data and your needs.