The variability that's shrinking when N increases is the variability of the sample mean, often expressed as standard error. Or, in other terms, the certainty of the veracity of the sample mean is increasing.
Imagine you run an experiment where you collect 3 men and 3 women and measure their heights. How certain are you that the mean heights of each group are the true mean of the separate populations of men and women? I should think that you wouldn't be very certain at all. You could easily collect new samples of 3 and find new means several inches from the first ones. Quite a few of the repeated experiments like this might even result in women being pronounced taller than men because the means would vary so much. With a low N you don't have much certainty in the mean from the sample and it varies a lot across samples.
Now imagine 10,000 observations in each group. It's going to be pretty hard to find new samples of 10,000 that have means that differ much from each other. They will be far less variable and you'll be more certain of their accuracy.
If you can accept this line of thinking then we can insert it into the calculations of your statistics as standard error. As you can see from it's equation, it's an estimation of a parameter, $\sigma$ (that should become more accurate as n increases) divided by a value that always increases with n, $\sqrt n$. That standard error is representing the variability of the means or effects in your calculations. The smaller it is, the more powerful your statistical test.
Here's a little simulation in R to demonstrate the relation between a standard error and the standard deviation of the means of many many replications of the initial experiment. In this case we'll start with a population mean of 100 and standard deviation of 15.
mu <- 100
s <- 50
n <- 5
nsim <- 10000 # number of simulations
# theoretical standard error
s / sqrt(n)
# simulation of experiment and the standard deviations of their means
y <- replicate( nsim, mean( rnorm(n, mu, s) ) )
Note how the final standard deviation is close to the theoretical standard error. By playing with the n variable here you can see the variability measure will get smaller as n increases.
[As an aside, kurtosis in the graphs isn't really changing (assuming they are normal distributions). Lowering the variance doesn't change the kurtosis but the distribution will look narrower. The only way to visually examine the kurtosis changes is put the distributions on the same scale.]