I too spent a few years with a similar desire to understand probability and statistics at a deeper level, but we might have been stuck on different things. Here's an explanation:

Suppose you have 37 numbers. Purchase a massless ruler and 37 identical weights. For each of your numbers, find the number on the ruler and glue a weight there. You now have a massless ruler with 37 weights glued onto it.

Now try to balance the ruler sideways on a spike sticking out of the ground. The mean of your numbers will be the point on the ruler where it balances.

Now spin the ruler on the spike. It's easy to speed up or slow down the spinning ruler if the weights are close together, but more force is required if the weights are far apart. The variance of your numbers is proportional to the amount the ruler resists changes to its angular velocity -- how hard you have to twist the ruler to make it spin, or to make it stop spinning.

"I'd like to understand this more deeply" is a thought that occurs to people at many levels of study, so this explanation could be too high or low. Where did my comment hit?