The truth is.
The proof is a mechanism to reach that consensus, by convincing other mathematicians of a specific truth. That is all it is.
There is a naive idea that a proof is a purely mechanical series of steps that provides access to truth. Last I checked, this isn't so for the vast majority of proofs in math. Such a proof would be way too tedious to construct or check by mathematicians. And if it isn't checkable, how do we know it is actually true?
Automated proofs are a subfield, and (again, last I checked) controversial because they can often not be checked by humans.
So for example, if the proof doesn't convince other mathematicians, then it's not a a proof.
Or it might convince other mathematicians and later turn out to be wrong after all.
For more on the practical aspects of math, I highly recommend The Mathematical Experience.
I read it in German:
https://www.amazon.com/Mathematical-Experience-Phillip-J-Dav... is a link to Amazon.
Fresh book recommendations delivered straight to your inbox every Thursday.