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A typical multirotor drone relies on vertical motion in order to fly. Think of the drone blades as fans. The air coming out of the spinning blades pushes downward on the ground, which in turn pushes the drone into the air. The faster the motors spin, the faster the drone will accelerate off the ground.
Once the drone is airborne it can hover, fly up/down, and left/right. When the drone is hovering the upward thrust of the motors is equal to the downward force of gravity, so the drone will not move up nor down. When the force of the motors exceeds the force excreted down on the drone by gravity, it will then climb higher and vice versa.
Since there are 4 motors, changing one or more of these motors speed, compared to the others will cause the drone to turn or move in a different direction. For example, if the drone is hovering (the upward thrust of the motors is equal to the downward force of gravity), and the 2 motors on the left are given more power, the drone will lift upward on the left side, and this increased power on the left side will start pushing the drone sideways right.
This is a simple summary of how a multi rotor drone flies. Fixed wing drones are different, and a bit more complicated to explain (think of a plane).