When we say a TM solves a problem, what does this mean?
To simplify, let’s only speak about decision problems.
Decision problems have yes or no answers.
For example; Is X prime? Is a decision problem.
We can re-formulate the following question to be a set membership problem.
We define a language L to be the set of all prime numbers, ie the set of all strings that would make the decision problem return yes. To clarify, if X is a prime number, then the answer to the above would be yes.
We now change our question from “is X prime” to “is X in L”.
Since L is a language, we can associate a TM which decides L. This means a Turing machine which will accept on all strings in L and reject on all strings not in L.
This TM can now answer the problem of “is X prime” , if it accepts then X is prime, if it rejects then X is not prime.
We can say a TM solves a problem, if it decides the corresponding language L to that problem.
minor note: WLOG I did not mention the encoding of X for simplicity. This however does not change the overall message