Can someone with understanding of satellite launches explain to a layman how the satellites get into their respective orbits when launched together like this?
Well, most of the satellites will want to be in about the same orbit - the fairly common sun-synchronous orbit, that passes over the same spot on earth every day. They probably want to spread the satellites out over the orbit, though.
I'm not sure if randomness would be enough to do this, but if not, they could probably control things a little bit, at least for the Dove satellites (88 of them). Since the Dove satellites can point themselves to take pictures, my guess is that they can also choose to point themselves either perpendicular or parallel to the orbital direction. Since there's a little bit of atmosphere left up there, over time that should be enough to intentionally spread the satellites out.
This is just a guess, but I know similar techniques have been used in the past on satellites that were out of fuel - for example, Skylab.
Either the last stage of the rocket is relit, the satellites can thrust themselves, or the small velocity they're released with over time will put the satellites in different orbits.
A relit final stage can put satellites in significantly different orbits. Satellite thrusters can do a little but only small orbital changes. And just releasing over a period of time will result in each sat being released on the order of 1 m/s which is enough to keep them separated but in the same orbit.
If a satellite has its own maneuvering capabilities, it can change its orbit (somewhat). These are mostly nanosats and cannot do that, so they'll stay in closely-related orbits. Spring-loaded separation from the launch vehicle will give them some difference, which will be magnified over time.
Some rideshare launches have multiple burns of the upper stage to put payloads in different orbits. The PSLV can do that, though I don't know if this one will.
