Interpreting the rule that dW squared equals dt

In stochastic calculus one writes $(dW_t)^2=dt$. What does this shorthand actually mean?

It encodes the quadratic variation of Brownian motion. Over a partition of $[0,t]$, $\sum(\Delta W)^2$ has mean $t$ and variance $2\sum(\Delta t)^2\to0$, so it converges to the deterministic value $t$. Each squared increment is random, but in aggregate they behave like $dt$. Hence in Ito's formula $(dW)^2$ is replaced by $dt$, while $dW\,dt$ and $(dt)^2$ are negligible.