It means that according to Cayman Airways it's standard industry practice not to be their responsibility for missing any of your connecting flights that were not booked with them to your original destination regardless of why the pjs don't fly. There are not that many places cayman airways flys to in the US. So most people are in this situation.
So, if you book your flight with any other airlines and cayman airways have mechanical problems or any other issues, you are really on your own for cancelling/rebooking/paying for your new flight. In this case, they will hand you the phone that's what they mean by assisting you in finding another flight. If you contact the Ministry of Tourism, you may get Cayman Airways to pay for your hotel and cab fare.
This is not a problem normally because this doesn't happen a lot with other airlines and there are usually other airlines that fly the same route if there is a problem.
The 737 flies direct from Miami to CB. But on the way back, it flies to GC first. They decided to move the entire time for the 737 in the late afternoon, but put people on a pj on the way back to GC. Fortunately, some of us bought our tickets after the schedule change occurred. Unfortunately, some bought their tickets before the change.