ArcMap vector exporters output all of the geometry for each feature intersected by the dataframe and draw a clipping path to hide the area of those features that would appear outside of the dataframe. Releasing the clipping path in Illustrator allows these features to show through. The graphic should print fine from Illustrator with the clipping path intact.
The fact that data outside the dataframe is still there is a feature of the software. Clip the data using geoprocessing tools to remove all data from outside the data frame, if needed.
An enhancement request, NIM002896, has been submitted to change this behavior.