Keplerian elements are the inputs to a standard mathematical model of spacecraft orbits. With the "keps", the correct time, and your station location, you can compute when the satellite will be in view and where to point your antennas. See also the tutorial on Keplerian elements and the explanation of the formats used in these files.
If you really want to know the mathematics behind the elements, see Spacetrack Report No. 3 from NORAD (1.1 megabytes in PDF format). Most amateur radio tracking programs use a simplified version of the simplest model described in this report.
AMSAT publishes Keplerian elements weekly. Here are the current bulletins:
You can receive these bulletins regularly by e-mail by subscribing to the KEPS mailing list.
Space Shuttle orbital data is available during missions.
You can find lots of Keplerian elements and related information on these other sites:
Updated 4 January 2005. Feedback to KB5MU.