For we don't give a direct account of what it takes for two truth-values to fall under the extension of "&" (and use some truth-table written on a board as an aid to express that).
Now, I'm not claiming that this process of introduction of the connective "&" is in a better position than Russell's truth-table on a blackboard story in the sense of being a better explanation of how this meaning attribution took place (though I actually believe it is).
The helpfulness of the truth-table becomes clearer, however, when we are dealing with more complicated propositions, such as "Today is not the First of December, and it is raining," with the second "Elementarsatz" (as Wittgenstein would say) now represented by r.
Wittgenstein knows unequivocally that tautologies are true under all conditions while contradictions are true under no conditions precisely because he has used the truth-table method to determine this much.
The objective of the procedure, which is an adaptation of the well-known short-cut (reductio ad absurdum) truth-table
method from extensional logic, is to show either that the formula to be examined can be decisively falsified or else that it cannot be falsified without an inconsistent assignment of truth values to the variables.
Most commentators believe Wittgenstein developed his truth-tables
from earlier hints in the works of Frege Gottlob and Bertrand Russell.