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 $N\to\infty$.