I don't recommend trying to work on the genset itself if not really good at that sort of thing. They are complex pieces of gear and take lots of training.
But the system you have will have a few points which can be checked by using a meter.
There will be a metal box where you switch the cord to get power from shore power to getting it from genset. If you take that cover off and check the wires inside, you may find one has come loose. This is best done with the genset off and not plugged, just as a safer move.
Assuming you've found the main output breaker on the genset and got it on as you mention?
That breaker and starting the generator should put power to the connections in the metal box and those can be checked for voltage there.
One of the things that can mess with our mind is a wire going bad inside the shore power cord, so give that some thought. Does the cord work when plugged to shore power, proving it is good?
Then it is also possible to plug the cord in but it not make good contact, to some wiggle and pushing it in/out a bit while watching for changes in the power can sometimes clear the easier things.
One handy indicator for good AC power is the microwave light. I use them as a handy indicator when I have questions about the AC side of it.
With the "manual transfer" setup, your system is pretty simple and less involved than with auto transfer, so it not one of the points I mentioned, the problem is likely out of the normal range for us to fix.
Click this snip to get better view!
If the power is good when plugged into shore power, all the points marked in green should be verified as good, leaving the main questions to be the wires in the box and possibly things like corroded plugs, etc.
The last big question is getting that power from the generator into the first metal box. We also have to consider the breaker on the genset might also be bad? Does it actually get power in one side and pass it out toward the RV?
If not getting it that far, I have to go for expert repair!