Why? Because they can. And besides, no one else had ever done it successfully.
I love amateur astronomers. Case in point: the San Antonio Astronomical Association members decided it would be fun to calculate when the International Space Station was overhead, find a big laser, and shoot that laser at the ISS when an astronaut was watching. Why? Because no one had ever done it. Because they could. Because they love space. Because they'll always have that picture you see up there, which was snapped by astronaut Don Pettit, to prove that they touched space, at least with some photons.
While it might seem easy to send a laser signal to the space station as it passes overhead, it's actually surprisingly difficult. Pettit explains:
This took a number of engineering calculations. Projected beam diameters (assuming the propagation of a Gaussian wave for the laser) and intensity at the target had to be calculated. Tracking space station's path as it streaked across the sky was another challenge. I used email to communicate with Robert Reeves, one of the association's members. Considering that it takes a day, maybe more, for a simple exchange of messages (on space station we receive email drops two to three times a day), the whole event took weeks to plan.The setup that eventually worked required precise timing, a one-watt blue laser, and a white spotlight.