If you are on a fixed gear, and you (attempt) to stop pedaling, you are likely to get pitched over the handle bars if you are going fast enough. Yes, the pedals do keep moving. That is where the name comes from. The gear is fixed to the hub, as opposed to what you think of as a "normal" bike where there is a ratchet system (or freewheel) in the hub. On a so called normal bike, when you stop pedaling, the wheel continues to spin, but the chain, and your pedals can stay still because the hub is "ratcheting." On a fixed gear however, when you stop pedaling, the wheel continues to spin, and because the gear, is directly attached to the hub without any kind of ratchet, the chain continues to spin which in turn continues to drive the pedals until the wheel comes to a stop.
It happens to almost everybody when the first start riding fixed. You forget that you are riding a fixed gear at some point, and you simply try to stop pedaling. If you aren't ready for it, the pedals will push back on your feet and can actually propel you over your handlebars. It is very common right after ascending a hill. You get to the top and want to sit back on the seat and stop pedaling for a little breather. You get quite the surprise when you try it.
Coincidentally, even though you probably think of a bike with a freewheel or cassette as a "normal bike" the first chain driven bikes were all fixed gears, including the first ones used in the tour de france. The riders had what are called "flip-flop" hubs with a different gear mounted on each side. When it came time to ride up a mountain, they would sit down on the side of the road, take the wheel off and flop it around for the easier gear, and get back on with the ride.