Both Hatef and Mayu are correct in their definitions of what "break" and "brake" mean.
However historically your friend is correct, there is a connection.
Trains, when they first came into use, had a habit of uncoupling the carriages attached to them, which was highly dangerous.
The solution was to attach a special carriage at the end, called the "Brake Van", note the spelling of "Brake".
If coaches became uncoupled this was called a "break", with the different spelling.
The "Brake Van", is such circumstances was able to stop the carriages still attached to it and the Engine was able to stop the carriages still attached to it. Both had "Brakes"
Why there is a difference is spelling I have not idea though.