I wear compression stockings occasionally, although I wore them more back when I was newly diagnosed with a DVT. Bascially, they keep the blood from pooling in your legs and force it back up--helping out with the body's battle with gravity and preventing clot formation by keeping the blood moving.

Doctors will also argue that they prevent PTS (post-thrombotic syndrome) for patients who have had DVTs, but that seems to be somewhat in doubt. They can, however, greatly improve pain and swelling in patients with DVT.

At this point, I will still wear them for long plane or car trips, just to keep the blood moving, although they aren't really necessary if you make a point to walk around every hour or so and if you stay hydrated.

I was considering wearing my running compression sleeves that I have for my class that I take twice a week. It's a 3.5 hour long class with a small 10 minute break mid way through. Maybe I should!

Veins don't work the same way arteries do. The heart only imparts pressure to the circulatory system on the arterial side. Once the blood passes through the capillaries, the venous blood returning to the heart is pumped there by your muscles squeezing the veins and pushing the blood along. Obviously the blood in the trunk veins gets pumped just by breathing. Blood from the brain and upper body above the heart is helped out by gravity. But the legs are a problem. Long veins where the blood has to be raised back up to the heart against gravity. That's why there are valves in the veins. Blood there can only go up if the valves work right. So what compression stockings do is to keep the blood moving back up the venous system of the legs so it doesn't "pool" down there and have a chance to clot. They also increase lymphatic flow, so you get less swelling from fluid buildup in the lower extremities.
So that's how they work. You should ask your physician which strength he thinks you need.