I find the intuitive explanation by Paul Wilmott particularly appealing.


Stochastic integrals
$$\int_0^t f(W_{\tau})\ dW_{\tau}$$
involve adding up an infinite number of random variables. There are several contributions to the sum: those that are *a sum of random variables* and those that are *a sum of the squares of random variables*, and then there are *higher-order terms*.

Add up a large number of independent random variables and the Central Limit Theorem kicks in, the end result being a normally distributed random variable. But what is its mean and standard deviation?

When we add up $N$ terms that are normal, each with a mean of $0$ and a standard deviation of $\sqrt{\delta t/N}$, we end up with another normal, with a mean of $0$ and a standard deviation of $\sqrt{\delta t}$. This is our $dW$. Notice how the $N$ disappears in the limit.

Then if we add up the $N$ *squares* of the same normal terms then we get something which is normally distributed with a mean of
$$N\left(\sqrt{\frac{\delta t}{N}}\right)^2=\delta t $$ 
and a standard deviation which is
$\delta t\sqrt{2/N}.$ 
This tends to zero as $N$ gets larger. In this limit we end up with, in a sense, our $dW^2_t=dt$, because *the randomness as measured by the standard deviation disappears leaving us just with the mean $dt$*. 

The higher-order terms have means and standard deviations that are too small, disappearing
rapidly in the limit as $N\to\infty$.