This is best answered by the manufacturer of the video transmitter/receiver unit, because the answer lies with the power being transmitted. Chances are the unit you purchased does not require an operators license. A good rule of thumb is if it is mass marketed and relatively inexpensive ($150 USD or less) it is such a low power transmission it isn't required to have an operators license. These units also have limited range and the quality varies from product to product.
Higher quality units that are upwards of $500 begin to dabble in higher power and possible license requirements. If you are just starting to experiment in the area the inexpensive units are a safe bet and easy on the wallet in case you decide you aren't really that into it.
If you want even better video quality for the money, you can skip the down-linked video route and set up a digital camera that record in flight. The benefit is better quality per $, but the drawback is you are "guessing" what you are shooting while in the air. This is less a problem than it sounds, a little practice and you will get more of an idea what you are shooting as you go. One of the most popular strap-on (non-downlinked) cameras right now seems to be:
http://www.rcgroups.com/forums/showthread.php?t=800950.
An example of a unit requiring a license is the set-up Dan Thompson has been using for several years now. The video is available on the MDA website at
http://www.mudduckaviation.com/video/Dan%20Thompson's%20coverage%20.mpg.
RCGroups has a great resource for aerial photography/videography at the following link:
http://www.rcgroups.com/aerial-photography-128/The beauty of the Ducks is that they are very capable of large space and heavy equipment requirements for the task if you opt for a more professional or maybe a quality but "larger/heavier" but less expensive set up. Most people are popping the tiny/lightweight units onto small planes because that is all the plane can handle, hence they become popular due to demand. Flying a Duck provides tons of space and great lift and capacity ability along with slow flight, exactly what you want when taking video or trying to capture a still shot: SLOW. If you have an old tape video camera, you could mount it in a Duck's belly easily to get very high quality video, but you lack the ability to pan and tilt if you so wish.
Greg