Part of the issue is that you are not subject to the principle of explosion. You can assert contradictory things without also asserting that 2+2=3, so you can be confident that you will never tell anyone that 2+2=3 without being confident that you will never contradict yourself. Formal systems using classical logic can’t do this: if they prove any contradiction at all, they also prove that 2+2=3, so proving that they don’t prove 2+2=3 is exactly the same thing as proving that they are perfectly consistent, which they can’t consistently do.
Part of the issue is that you are not subject to the principle of explosion. You can assert contradictory things without also asserting that 2+2=3, so you can be confident that you will never tell anyone that 2+2=3 without being confident that you will never contradict yourself. Formal systems using classical logic can’t do this: if they prove any contradiction at all, they also prove that 2+2=3, so proving that they don’t prove 2+2=3 is exactly the same thing as proving that they are perfectly consistent, which they can’t consistently do.
Unless I’m missing something, Löb’s theorem is still a theorem of minimal logic, which does not have the principle of explosion.