quasars - What is projected separation and how can I make sense of its unit (h^-1 kpc)?

It means the separation between the two objects if they were both at the same distance away. This separation is found by multiplying the angular separation (in radians) by the distance to the objects.

Often, the distance to extragalactic objects depends on the assumed value of the Hubble parameter. In this case, $h$ is a dimensionless number corresponding to $h=H_0/100$, where $H_0$ is the Hubble parameter in km/s per Mpc.

The current best value of $H_0$ is about 70 km/s per Mpc, so $hsimeq 0.7$.