This is how I understand it (and I've only dove these a few times). There is no anchoring - some of these rigs are 700+ feet deep making that unpractical (as well as all the cables on the sea floor). The dive boat will drop the divers off fairly close to the rigs. They, as I understand it, are in constant communication with the rig personnel (as I've been on a boat that was waived off do to unsafe conditions above). The rig has the final say if the divers can jump or not. It's very quick - you get off the boat quickly, swim to a spot underneath the platform, and descend whilst in-between the legs. There can be considerable surge on the outside of the legs so it's best to get inside as fast as possible. From there, you descend. The top 20' or so is pretty much barnacles. The rest of the way down is glorious life. There are cross beams every 55' (I believe that's the depth).
In the case of the PacStar, there's a chase boat that circles the rig ensuring the divers are safe and that everyone's operating in a safe manner.
Pickup is done pretty much the same. You surface beneath the platform, signal the dive boat and the captain will tell you what to do - usually swim out to a spot where they will pick you up. Again, it's done quickly and efficiently. If necessary, the boat will throw out a line and help you come in.
The big point is that it's a quick off and on - and the PacStar does it beautifully. I am always amazed when watching the captains skills in getting in close and tight to the rig.
I really have no idea what rigs in SoCal are active and which are not, but most definitely people man the Eureka. We were about ready to launch (that being right about to do a giant stride) when the rig called to let us know they needed to do emergency work on the top platform and worried about dropping tools and other items on the heads of divers. So, we went to Catalina.
It's quite an adventure and an amazing dive if you can do it. But it's a dive that must be done with precision and safety in mind - for everyone.