What is the airspeed velocity of an unladen swallow?
In general, the method of getting the airspeed of any bird is pretty much the same:


Measure the ground speed of the bird.
Measure the wind speed in the area.
Calculate the airspeed of the bird from these two values.


The following excerpt from Air speeds of migrating birds observed by ornithodolite and compared with predictions from flight theory by C. J. Pennycuick et al. published by the Royal Society details one such method:


  We measured the bird's air speed, for comparison with the predictions of theory, in two stages. First, we measured the ground speed vector, consisting of the bird's speed relative to the observer's position on the shore and its track direction, i.e. the direction in which the bird was moving over the ground, measured clockwise from true north. We also measured the wind vector consisting of the wind speed and the direction from which the wind was blowing, and then obtained the air speed vector (air speed and heading direction) by vector subtraction of the wind vector from the ground speed vector 


Other researchers have used  Doppler radar for the same purpose and in case of high flying birds, one can simply fly along with the birds at the same speed and measure the airspeed.