Understanding of Turing's Answer to the Entscheidungsproblem
What Gödel showed is that there is no finite first-order axiomatization of the theory of the natural numbers. What this means is that if you give a finite list of first-order axioms and schemas, then the corresponding proof system won't be able to prove all true statements about the natural numbers. For example, it won't be able to prove its own consistency (this is Gödel's second incompleteness theorem). (A proof system is consistent if it doesn't prove a statement and its converse.)

Turing's machine, when applied to any given finite first-order proof system $\Pi$, will thus be able to prove neither the statement that $\Pi$ is consistent nor its negation.

Of course, this is not a proof that the Entscheidungsproblem itself is unsolvable; perhaps there is an altogether different approach that does work. Turing was able to show that, in fact, no (computable) approach works.