How do convection currents cause wind?
The best example is a sea breeze.
Convection occurs when the Earth is heated unevenly. The air over the Earth will heat according to how fast the Earth under it heats. That part of the atmosphere that heats faster will expand. This relationship between heat and volume is Charles Law.
Since the volume increases the pressure will drop. This is due to the fact that air masses do not readily mix, so the amount of air remains more or less constant and the volume increases. The relationship between volume and pressure is called Boyle's Law.
So the warmer air has a drop in pressure and lower pressure than the surrounding air. The difference in pressure over a distance is called a pressure gradient, and this creates a force called the pressure gradient force (simply that air wants to move from an area of high pressure to an area of low pressure), which is the driving force behind wind.
This is best illustrated in a diagram of a sea breeze.
So the warm air over the land heats, then expands, and the pressure drops causing the air to rise. Over the water the air is cooler and the air does not expand and the pressure is higher than the land so the air over the water moves to the land. The processes that occur aloft are not really important in understanding what is happening at the surface.
So in short, convection (rising air due to heat) lowers surface pressure creating a pressure gradient that tries to correct itself by having air move from higher pressure to the lower pressure area created by the rising air. This correction is wind.