Your double pole breaker has two connections to the two bus bars in the barn's load center. It will also have two screw connections on the outboard side to connect to wires. Both of these screw connections are 120V. If using conventional building wire, you would have a black wire to one screw connection and a red wire to the other screw connection. The white neutral wire connects to the neutral bus bar in the load center and a green wire connects to somewhere in that same load center.
The receptacle is wired with one black wire to a positive post, a red wire to the other positive post, the white goes to the neutral and the green wire goes to the ground terminal. There is plenty of information and diagrams out on the web for wiring receptacles and breakers into load centers. Just make sure you are using safe techniques while you work, If you can disconnect power to the barn's load center while you work, that will be the best.
FWIW, I installed 50 amp service in my TT. It is a good feeling to have plenty of power available and that the power inside the RV is well distributed. I've never tripped a breaker, either in the TT or at the load center.
Good luck with your new receptacle.