The tension of the string pulls the pendulum in and keeps it moving along path of a circle. The highest tension occurs when pendulum moves at its fastest - through the bottom of its swing.

At this point the string must oppose the full force of gravity AND keep the pendulum to its circular path (if the tension was only enough to cancel out the force of gravity the pendulum bob would continue in a straight line).

The string gets to apply this force (its tension) to the pendulum bob only by being stretched a bit more as the pendulum moves away - the previous tension not being enough to maintain the circular movement.
The "cord" of any pendulum - even if it's a solid piece of metal - will stretch more  (even if just a very little) a the bottom of the pendulums swing. The stiffer the "cord" the less the stretch will be.

On a swing