The answer is noone is going to count every single one. In the video they said they already know the amount. Which means they know approximately how many per case (on average) and how many cases they put in.
Either that or they filled the case, emptied it and weighed the candy corn, then used average candy corn/pound to figure the number.
Watch the video carefully and you'll have a pretty good idea of how many cases they put in.
As previously mentioned however, there are structural elements within the case which limit how much candy may be added. Even if you knew exactly how much candy corn will fit in those dimensions, you have to shave some off from the total number.
Its still a total guess, just like any contest should be.
Although if you actually owned the case you could determine the dimensions of each compartment and have a fairly good idea, but still a guess really.