Exterior angle is the angle for the chosen side of a shape where a line is extended from the next side. In simple terms exterior angle is the angle between one side of a triangle and the extension of an adjacent side.
The sum of opposite interior angles will be equal to an exterior angle of a triangle. If an equivalent angle is taken at the vertex then the exterior angles will add to 360$^{\circ}$.
Interior Angle    

Statement: The measure of an exterior angle of a triangle will be equal to the sum of the measures of two remote interior angles.
Exterior Angles Theorem
The angles, $\angle$ ABC and $\angle$ BCE are the interior angles of the $\triangle$ABC.

To Prove: $\measuredangle$ BCE = $\measuredangle$ A + $\measuredangle$ B

The exterior angle $\angle$BCE is supplementary to the adjacent interior angle $\angle$ACB.

As the sum of the angles of a triangle = 180$^o$
$\measuredangle$ A + $\measuredangle$ B +$\measuredangle$ ACB = 180
$\Rightarrow$  $\measuredangle$ A + $\measuredangle$ B = 180 - $\measuredangle$ ACB   ............(1)

Again $\measuredangle$ BCE + $\measuredangle$ ACB = 180  (Linear Pair)

or $\measuredangle$ACB = 180 - $\measuredangle$BCE     ...................(2)

Substitute equation (2) in (1), we get

$\Rightarrow$  $\measuredangle$ A + $\measuredangle$ B = 180 - (180 - $\measuredangle$BCE)

= $\measuredangle$BCE

=> $\measuredangle$ BCE = $\measuredangle$ A + $\measuredangle$ B.

Hence proved.