In the simplified semantics for logical languages, meaning is "proxied" by the truth conditions for sentences.
There is a long tradition in modern logic regarding "extensionality". From Frege's Begriffsschrift onwards, mathematical logic has been concerned mainly with truth-functional contexts.
According to Frege, the reference (Bedeutung) of a proposition is its truth-value, either the True or the False. For Frege, complete propositions, like names, have objects as their Bedeutungen, and in particular, the truth-values the True or the False. In this way, he is able to transcribe sentential connectives such as “and” and “or,” etc., as truth functions in the strictest sense — functions that take truth-values as argument and yield truth-values as value.
According to Alfred North Whitehead & Bertrand Russell, Principia Mathematica to *56 (2nd ed - 1927), page 115 :
It is obvious that two propositions are equivalent when, and only when, both are true or both are false. Following Frege, we shall call the truthvalue of a proposition truth if it is true, and falsehood if it is false. Thus two propositions are equivalent when they have the same truth-value.
We shall give the name of a truth-function to a function f(p) whose argument is a proposition, and whose truth-value depends only upon the truth-value of its argument. All the functions of propositions with which we shall be specially concerned will be truth-functions, i.e. we shall have
The reason of this is, that the functions of propositions with which we deal are all built up by means of the primitive ideas. But it is not a universal characteristic of functions of propositions to be truth-functions. For example, "A believes p" may be true for one true value of p and false for another.
In "modern" terms, this is the so-called Replacement theorem; see S.C.Kleene, Introduction to Metamathematics (1952), page 116 :
If A and B are formulas, C_A is a formula constructed from a specified occurnce of A using only the [truth-functional connectives], and C_B results from C_A by replacing this occurrence of A by B, then A ↔ B ⊢ C_A ↔ C_B.
In a "universe" of extensional contexts, like that considered by the simplified semantics of logical languages, it is natural to equate the condition for two equivalent formulae having the same truth conditions with that of expressing "the same thing", i.e. "having the same meaning".