Follow-through is important in all sports. It is especially important in archery because everything that is crucial to a good shot happens in a millisecond. Your follow-through shows you what you were doing during that critical millisecond.
A good follow-through indicates you stayed with the shot and that all of your muscles stayed relaxed until the conclusion of the shot. A poor follow-through shows that you didn't stay with the shot.
You don't have to do anything extra at the end of the shot -- just continue what you were doing before the shot until the arrow hits the target. If you wear a bow sling, you don't have to grab the bow, so your hand stays relaxed as it was before the shot.
Some archers believe perfect follow-through means holding the bow exactly where it was during the aiming process. I disagree. A big part of what was holding the bow up was force on the string. Once that force is gone, the bow will naturally fall forward and down slightly.
For the bow to remain exactly where it was during the aiming process, the deltoid muscle of the shoulder has to spring into action just as you release the arrow. You don't want any muscles jumping into action during that critical moment. Just relax and follow through.