This distance is the base of what astronomers call the fundamental distance ladder which eventually lets them determine the distances to nearby stars, star clusters, galaxies and quasars. To measure the 'Astronomical Unit' you determine the velocity of the Earth in its orbit , 29.79 kilometers per second, and then knowing that it takes the earth 31,556925.9747 seconds to go once around the Sun along the circumference of its orbit ( distance = 2 x pi x radius squared) you can use the orbital speed to determine the mean radius of the of the orbit. The velocity of the Earth can be deduced from measurements of a phenomenon called 'stellar aberration'.
Alternately, you determine the orbit of an asteroid such as Eros and at a given date calculate its distance in units of the AU. Then you bounce a radar pulse off of the asteroid to get its distance in kilometers. You then know how many kilometers the distance of the asteroid in AUs corresponds to.
The current value for the Astronomical Unit is 149.598 million kilometers.