Determine the output signal given the input and inpulse response
I'll answer your questions in reverse order, because the answers will make more sense that way.


  How did they get h(t) into the frequency domain?


By looking the function up in a table of Fourier transforms, most likely. The same way you'd normally do a Fourier transform:



  How were they able to determine the h(t) is a low pass filter?


By noticing that there exists some frequency \$\omega_x\$ for which \$H(\omega) = 0\$ for all \$\omega&gt;\omega_x\$.