A propeller is a wing that goes round
How a spinning blade makes thrust the same way a wing makes lift.
A propeller blade is a wing bent into a circle: rather than moving forward in a straight line to tilt air backward and down for lift, it spins in place and tilts air backward for thrust, using precisely the same trick a wing uses, just aimed in a different direction.
The physics of thrust from a spinning blade
Cut a propeller blade at any point along its length and the cross-section looks like a small wing, curved on one face and flatter on the other, and set at an angle to the direction it is moving through the air. As the blade spins, that little wing-shaped section meets the air at an angle, exactly as a wing does when it is tilted nose up into an airflow, and it deflects the air it meets backward and slightly to one side. Deflecting that air backward means the blade section is pushed forward in reaction, by the same law that pushes a wing up when it deflects air down. Adding up that forward push from every section along every blade, spinning around many times each second, gives a total that is thrust, enough to drag an aircraft, or a boat, forward through whatever fluid it is spinning in.
The difference between a wing and a propeller is really only in how the two arrive at their working angle. A wing gets its angle of attack from the pilot pitching the whole aircraft up or down, tilting the entire surface into the airflow at once. A propeller blade gets the same effect from its own twisted shape, set once by whoever designed it, and from the rotation itself, which carries every part of the blade steadily into new air, so that a fixed shape can go on generating thrust indefinitely without ever needing to change its angle relative to the aircraft.
Where you have already felt this
Turning a wood screw with a screwdriver gives a feel for why a propeller blade is twisted rather than flat along its length. A screw's thread is a ramp wound around a shaft, and turning the shaft makes the ramp pull the screw forward into the wood a fixed distance for every full turn, however fast or slow the turning happens. Early propellers were called airscrews for exactly this reason, since a blade set at a constant angle along its whole length would, in principle, pull an aircraft forward a fixed distance for each revolution, the same way a screw thread does. Air is far less obliging than wood, though, since it slips and compresses instead of gripping the blade the way wood grips a screw thread, and how much it slips depends on how fast a given point on the blade happens to be moving.
The one number worth remembering
How fast any point along a spinning propeller blade is moving through the air depends on only two things multiplied together: how quickly the propeller is turning, and how far that particular point sits from the centre of the hub. A point two-thirds of the way out along the blade covers twice the distance in one full turn that a point one-third of the way out covers, simply because it is travelling around a bigger circle in the same single revolution. This is why propeller blades are twisted, flatter and less angled near the tip, steeper and more angled near the root, so that every section along the blade, despite moving at a different speed, still meets the air at roughly the same efficient angle.
What follows from this
Because the outer part of a blade is always moving faster through the air than the inner part, it is the tip that runs into trouble first as a propeller spins faster or an aircraft flies faster, since the tip's speed adds to the aircraft's forward speed while the root's does not by nearly as much. Pushing a propeller too hard lets the tips reach speeds where air stops behaving the way the rest of the blade assumes it will, which is one of the reasons propeller-driven aircraft have a practical ceiling on how fast they can usefully spin their blades, however much power is available to spin them faster still. It is also why larger propellers are often geared to turn more slowly than the engine driving them, trading rotation speed for blade length so that the tips stay well short of the speeds where the whole picture of a wing simply going round begins to break down.