In modern terms, you can identify sequences and automata: a sequence is a current state, plus a transition function giving you the next element of the sequence and the next state. However, if every function is tracked by a computable process, then the transition function must also tracked by a computable process, and so every infinite sequence must be generated by a computable process.

This identification is precisely what choice sequences are used to rule out. If a sequence is genuinely arbitrary, then you cannot describe it up front with a finite amount of data and a finite rule for updating that data. Instead, you have to imagine a mathematician adding a new axiom at each time step about the next element of a choice sequence, and requiring that any constructions they do works uniformly regardless of what axioms may be chosen in the future.

In this picture, the total amount of axiomatic data is actually going up over time, and this is actually important for the results Brouwer proved. In particular, without the axioms for choice sequences, you cannot derive the strong counterexamples, because you need classically-false axioms to prove classically-false theorems (like the continuity of all real-valued functions).