You simply just try all possible ways to split the string. The non-determinism just means that we assume that we can guess the correct split. This can however be simulated in a deterministic way by trying all solutions. Since we are talking about decidable languages, there is a decider that will never go into an endless loop on any input.
Here is a deterministic proof. Assume we have a decidable language A, then there exists a decider $M_A$. From this $M_A$, you can now construct the following decider for any language in $A^*$:
On input w:
If w is the empty string, accept
For all possible splits, $w = x_1x_2...x_n$: if M_A accepts all the strings $x_1, x_2 ... x_n$, then accept
If no split was accepted, reject.
Whether a non-deterministic TM can be simulated in an efficient way by a deterministic TM is an open question. That is, does there for any problem we can solve efficiently by non-deterministic TM always exists a deterministic TM that wont have to try all possible "guesses" the non-deterministic TM does?