We know that any tangent forms 90 degree angle with the radius of the circle.
In this example, let the tangent from an ext point P meets the radius of the circle with center O at Q. So, angle PQO forms 90 degree (PQ is the tangent and OQ is the radius. Now, joining point O and P we get a Triangle OPQ (angle OPQ=90degree,Thus, It is an right angled triangle) with its hypotenuse OP
(Side opp to 90 degree angle is hypotenuse. In this case, OP is opp to 90 degree angle PQO and thus OP is the hypotenuse of the triangle PQO). And we know that the hypotenuse is the greatest side of the triangle. Therefore OP is the greatest side of the triangle OPQ and thus OP is greater than the tangent from the externel point P.