I find the intuitive explanation by [Paul Wilmott][1] particularly appealing.


Fix a small $h>0$. The stochastic integral
$$\int_0^{h} f(W(t))\ dW(t)=\lim\limits_{N\to\infty}\sum\limits_{j=1}^{N}
f\left(W(t_{j-1})\right)\left(W(t_{j})-W({t_{j-1}})\right),\quad t_j= h\frac{j}{N},$$
involves adding up an infinite number of random variables. Let's substitute every term $f\left(W(t_{j-1})\right)$ with its formal Taylor expansion. Then 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. Let's calculate 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 $\sqrt{h/N}$, we end up with another normal, with a mean of $0$ and a standard deviation of $\sqrt{h}$. This is our $dW$. Notice how the $N$ disappears in the limit.

Now, if we add up the $N$ *squares* of the same normal terms then we get something which is normally distributed with a mean of
$$N\left(\sqrt{\frac{h}{N}}\right)^2=h$$ 
and a standard deviation which is
$h\sqrt{2/N}.$ 
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 $N\to\infty$. 


  [1]: http://www.amazon.co.uk/Paul-Wilmott-Quantitative-Finance-2nd/dp/0470018704