How do wireless chargers work?

Electrical current through a wire creates a magnetic field directed tangential to the direction of the current (in a circular motion around the circumference of the wire). So, when you coil the wire into a circle, this creates a magnetic field in the direction perpendicular to the circular cross-section of this coil (think of a donut of wire sitting on a table, the magnetic field would be directed upward or downward through the hole of the donut).

Now, if you take a second coil of wire and place it on top of the first coil, the magnetic field from the first coil will cause a flow of current in the second coil. This is due to the reverse of how you generated the magnetic field.

The "first coil" is your wireless charger, and the "second coil" is inside your phone, connected to the battery. The current generated in the second coil charges your phone's battery.