An intuitive explanation given by Roger, who pointed out that delta hedge insulates you from only small changes in stock (or another underlier).
Say, you use 50 shares to hedge an ATM option position with delta hedge ratio = 50%. Suppose your hedge is not dynamic so you don’t adjust the hedge, until price moves so much the option delta becomes close to 100%. Now a $1 move in your option is offset by $0.50 change in the stock position – insufficient hedge.
The other scenario Roger didn’t mention is, the option becomes deeply OTM, so delta becomes 1%. Now a $0.01 change in your option position is offset by $0.50 of the stock position – overhedged.