Intuition behind the Ito correction term

Explain why Ito's lemma takes the form it does.

Taylor expansion where the second-order term survives. Expanding $f(t,X)$ gives $f_t\,dt+f_x\,dX+\tfrac12f_{xx}(dX)^2+\dots$. In ordinary calculus $(dX)^2$ is negligible, but for $dX=\mu\,dt+\sigma\,dW$ the increment $dW$ is of size $\sqrt{dt}$, so $(dX)^2=\sigma^2dt$ is first order. Higher terms vanish, leaving $df=(f_t+\mu f_x+\tfrac12\sigma^2f_{xx})dt+\sigma f_x\,dW$; convexity of $f$ creates a drift.