The databases are the .edb files, which are just text files. The format is documented under
Objects->Load/Delete->Help->File format.

However, that just gives you the equatorial coordinates of the objects. The rise/set times are computed by XEphem using additional info such as time, earth orbit models and topocentric geometry. You can dig out the algorithms if you like by starting with riset_cir.c.

You might also look into xephemdbd, the batch-oriented version of XEphem. It uses the same databases and algorithms but uses text streams for input and output, no GUI at all. Look in tools/xephemdbd just beneath the XEphem source directory for comlpete details. It can, among other things, print rise and set times.

Elwood