The accepted answer explains how wings can help reduce the required energy for flying by pushing a lot of air slightly, in place of forcing a little air strongly.
The idea that the required thrust can even be made less than the weight of the body is intriguing, and this is not possible without wings. The following discussion attempts to explain it.
The formula $F_{drag} \propto v^2$ holds the answer to the question what wings can do but engines can't.
It tells us that in air, velocity of a body can be used to generate a force that, in turn, can be used to fight the weight of the body.
And since velocity, a component of momentum, is conserved, it should be theoretically possible to have a flying body with no external power needed!
So, we need a mechanism to generate resistance, and redirect it (or at least some of it) opposite to gravity.
Both of these things are done by wings. In very simplistic terms (not considering Bernoulli's principle and friction etc.), the picture below roughly shows what wings do.
So, I would rather ask the question - what does an engine do that wings can't!
Both of these things cannot be done in any significant way by the engine - generating resistance and redirecting it. Yes, engine can generate thrust, and a lot of it, so much so that we won't even need air resistance to generate lift, and that's what helicopters do. But, if we take the help of air resistance, we may not need thrust at all, or at least we'll save a lot on thrust.
The wings do generate resistance, but unfortunately they can't redirect all of it vertically - some it is left out to act horizontally - as drag - opposite to velocity. This is where engines are needed. They just counter the drag that is less than the weight of the aircraft.
So, it's wings that do the flying given an initial velocity, and engines just help wings maintain that initial velocity. The wings are always falling in a horizontal direction onto the air in front of them, and due to the special angle of attack, in spite of the horizontal fall, some of the force of resistance is redirected vertically upward, and that does the trick.