Let's start by looking at how many variations there are if we had six symbols and a 1-sided "die". Pretty obviously, there are six.
Now, how about for a two sided die? We just established the number of possible variations for each side individually, but how can we combine them together?
Well, notice that if we lock in symbol A on the first side, and we let second side vary between any of the six six, we would have six possibilities again. If the first side had symbol B on it, we would again have six possibilities, and so on for the other choices for the first side. That is, for any particular symbol on side 1, we have six possibilities for side 2, and since we have six possibilities for side 1, that gives us 6*6, or 62 possibilities overall for two sides.
With that, we have a nice way to extend things to any number of sides (and any number of symbols). In general, if you have s symbols and n sides, the total number of possibilities is sn.
In our case we have 6 symbols and 6 sides, so the number of different variations is 66 = 46656.