If the system of axioms is consistent, it means that you cannot prove both a result and its contrary. If the system is complete, then you can also prove any assertion to be true or false. It follows that a mechanical procedure would ensure that all mathematical assertions can be decided mechanically. In principle, if the set of axioms is consistent and complete, you can decide whether any theorem is true or false. Such a procedure means that a mathematician no longer needs ingenuity or inspiration to prove theorems.
Mathematics becomes mechanical.
Of course, mathematics is not like that. But I think that the Turing version of the theorem is more fundamental and easier to understand. He showed that there is no mechanical procedure for deciding whether an arbitrary program will ever finish its computation and halt. To show that the so-called halting problem can never be solved, we set the program running on a Turing machine, which is a mathematical idealisation of a digital computer with no time limit. Turing showed that there is no set of instructions that you can give the computer, no algorithm, that will decide if a program will ever halt.