Let’s take a quadrilateral as shown in figure. By dividing it diagonally, we can see that it breaks into two triangles.

We know that sum of three angles of any triangle is equal to $180°.$
Hence, $\angle QPR + \angle Q + \angle QRP = 180° and \angle SPR + \angle S + \angle PRS = 180°.$
Now to get the sum of four angles of quadrilateral,
$\angle P + \angle Q + \angle R + \angle S = (\angle QPR + \angle SPR) + \angle Q + (\angle QRP + \angle PRS) + \angle S$
$= (\angle QPR + \angle Q + \angle QRP) + (\angle SPR + \angle S + \angle PRS)$
$= 180° + 180°$
$= 360°$