"This conjecture says that 47,176,870 is the maximum number of steps that a 5-state Turing machine can run before halting (starting from all-0 memory tape)."
So, we can run up to this number and either the machine will stop - and we'll go to the next one until exhaust the whole set - or the machine won't stop, and we'll refute the conjecture.
If you simulate a Turing machine up to that many steps, and find it hasn't halted yet, then the conjecture is refuted if you know the machine's execution will eventually terminate -- but famously, there's no algorithm that can reliably tell you whether or not a given TM halts.
If it was just about the maximum number of steps that could be executed by a 5-state Turing machine, it would be trivially false, because it's easy to construct a Turing machine that runs forever.
"This conjecture says that if a 5-state Turing machine runs for more than 47,176,870 steps without halting then it will never halt (starting from all-0 memory tape)."
No, you do not refute the conjecture if you run up to this number and a machine does not stop. Finding a machine that runs for more than 47,176,870 steps is easy - there are plenty of machines that run forever. The trick is that it needs to stop.