It is better to think of it as a "saving" rather than as a"surplus".
Also, it is better understood if we imagine heterogeneous consumers for whom a threshold price exists, a "maximum willingness to pay". Then at a given price level, some consumers are willing to buy the product and are expressing their demand for the product, while others are out of the market, because the price is still above their maximum willingness to pay.
If the price falls more towards the equilibrium, they enter the market.
But the ones already in the market would be willing to buy the good at a higher price. With the added consumers/higher output, they buy the good in a lower price than their maximum willingness to pay, and in that sense, they "save".
Imagine that the market demand function is build gradually not by the same persons increasing their quantity demanded, but by the addition of new consumers (this is easier imagined if we think of durable goods of which most consumers will usually buy just one unit or not at all).