Why are larger samples required to estimate higher moments than when estimating the mean? The Wikipedia article on moments states that "The higher the moment, the harder it is to estimate, in the sense that larger samples are required in order to obtain estimates of similar quality". Why is this so? And what does "quality" mean here? 
 A: To answer your second question first - "quality" means "accuracy", and accuracy can be defined in many ways, hence the lack of mathematical precision in the definition.
Higher moments are harder to estimate in many situations, easier in others.  If the probability distribution of the data is such that the mean equals 0 and all the data lies in $(-1, 1)$, the higher moments will be more accurately estimated, as they will in general converge to zero as the index of the moment goes to infinity.  Estimating the 101st moment of a variate that is distributed uniformly over (-0.5, 0.5) with high accuracy is really easy:
x1 <- x101 <- rep(0, 10000)
for (i in 1:length(x)) {
  u <- runif(5)-0.5
  x1[i] <- mean(u)
  x101[i] <- mean((u-x1[i])^101)
}
> sqrt(mean(x1*x1))
[1] 0.1289411
> sqrt(mean(x101*x101))
[1] 1.880887e-16

The RMSE of the estimate of the mean based on a sample size of 5 is $0.129$, give or take a little sampling error over our 10,000 samples, but the RMSE of the 101st moment is $1.9\text{x}10^{-16}$, far smaller.
However, in cases where there is substantial probability of values somewhat greater than 1, the story changes.  Now, because we are raising the larger sample values (those $>1$) to higher powers, they become bigger, rather than smaller.  Consider the same experiment, but with the variate distributed uniformly over (-5, 5) (skipping the trivial rewrite of the code):
> sqrt(mean(x1*x1))
[1] 1.290788
> sqrt(mean(x101*x101))
[1] 4.029381e+85

You can imagine that it will take a LOT of data to get that $4\text{x}10^{85}$ RMSE for the 101th power down to roughly the same accuracy as the estimate of the first moment (1.3).  Here's what happens when we increase the sample size from 5 to 5000:
> sqrt(mean(x101*x101))
[1] 4.596604e+68

A big reduction, to be sure, but still a long way to go.
As alluded to above, the reason for this is that when we calculate the sample estimates of higher level moments (by calculating the corresponding moment of the sample data), we are raising the observed numbers to higher and higher powers.  When they are $>1$, this makes them larger and larger.  Consequently, the numerator of the moment calculation gets larger and larger, so you need a larger denominator (which is the sample size) to compensate.
Note also that if you make assumptions about the distribution of the data, the Wikipedia statement need not hold.  For example, if we assume the data is Normally distributed, our "estimate" of all the odd moments will equal 0 regardless of sample size or how large the moment is.  
