ELI5: How does philosophy answer mathematical questions?

For context, I watched Veritasium's video about math's fundamental flaw and I could not understand how does Gödel's Incomplete Theorem prove that math not complete and is inconsistent? Same goes with Turing's halting problem because I don't understand how does the thought experiment about a computer could be used as prove that math is not decidable (no algorithm that can always determine if a statement is derivable from the axiom, according to Veritasium)

I had thought that a math formula/equation needed to be discovered/published to show that math is incomplete, inconsistent, and solvable/decidable since the original questions were about math.

How do mathematicians know these philosophy (such as self reference paradox) is answering whether mathematics is complete, consistent, decidable, or not?