Why there is a $\frac{1}{2}$ in the distance formula $d=\frac{1}{2}at^2$?

It is exactly because we have a factor of $\frac 1 2$ in the area formula of a triangle. To understand what I'm saying, consider what is the $v(t)$ graph of a particle under constant acceleration.

Some say, a good plot is worth a million words! :)