In essence, a bottom drain sits at the bottom of the pond and acts as the point of entry for water/debris into the filter. Gunk is pulled into it and then lands into your filter and the water then circulates through the pump and comes out clean on the other end back into your pond via a waterfall or whatever way you use to get water back in (hose, spitter, etc).
Proper bottom drains are dug under the pond so that water/debris flows to filter. This means you must cut a hole in your liner, so that you can fit the bottom drain into it. You seal the bottom drain properly into the liner so that no leaking occurs and voila!
There are also folks like me, who retrofitted an existing pond built by the previous owner (aka: fool) who did not build the pond with any filtration in mind. Therefore, we purchase what are called retro-fit bottom drains [also known as vacuum drains] that serve a very similar purpose except that they sit on top of your liner on the bottom of your pond floor. You still have to cut a hole in the side of your liner to get the piping through. Though my retro-drain is wonderful and has made a world of difference in my pond filtration--it is not as efficient as a true BD. When I build my second pond from scratch, it will have a proper bottom drain.
With either one, your crap goes into your filter and then you dump the waste from there. In my case, my filter is a DIY barrel--so I just open up the waste spigot and the debris goes out onto my lawn when I choose to flush it or do a partial water change.
Of course, there are plenty of folks here that do not have BD's at all as per their personal choice and have wonderfully clear water. I have had my pond without a BD and with one, and for me there is no comparison in terms of ease of keeping the pond clean that it has been less work and is easier with a BD.