Think of countable infinities as a line of stones along a very wide, infinitely long beach, set along at hundred-meter increments. Each stone represents a whole number, you can walk past some and make definite progress.
Think of uncountable infinities as the sand along that beach. Between each stone you can find grains at the halfway point, quarter-way point, every fraction we can write.

While the stones go the entire length of the beach, as far along as the sand does, there are infinitely more grains of sand. For the countable infinities, you could confidently say "I have passed five stones." With uncountable infinities, it is as if you must first count a sand grain, but you find that there's an infinite number of sand grains between the starting line and whatever grain you're trying to get to, and an infinite number of sand grains between *those* sand grains.

To bring the idea of uncountable infinities out of the metaphor, consider counting every fraction between `0` and `1`. You could count by tenths, `1/10`, then `2/10`, and so on. However, before you get to `1/10`, you neglect the hundredths in between `0` and `1/10`. But between `0` and `1/100` is `1/1000`. This conundrum continues down infinitely. These sets of numbers are called uncountable because you cannot count them in the same way you can count the natural numbers (like the stones on the beach). That is what is meant when it is said some infinities are larger than others, categorical distinctions such as these.
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