If you have access to an Advanced ArcGIS license, there is a handy geoprocessing tool that will accomplish what you need.
ArcGIS Help (10.2, 10.2.1, and 10.2.2) - Tabulate Intersection (Analysis)
Note that you will want to fill out the Class Fields (optional) and Sum Fields (optional) settings.
If you don't have access to an Advanced license, it could get a bit more involved depending upon your data. One could run an Intersect (Analysis), then in Excel do a Pivot Table. The tricky part will be if you end up with multiple polygons of the same ID in the same Fishnet, as the multiple polygon with the same ID will be be listed separately and not aggregated.
Chris Donohue, GISP