Why does the sss postulate work?
According to a universally accepted system of axioms of Geometry developed by Hilbert, only side-angle-side (SAS) is an axiom.
Two others, ASA and SSS are theorems requiring proofs.
Strictly speaking, side-side-side (SSS) congruence of triangles, as well as angle-side-angle (ASA) are theorems (statements we should prove).
Only side-angle-side (SAS) is accepted in the Hilbert system as an axiom.
Postulates and axioms are equivalent terms and mean the same - they are the statements that we accept without proof.
IMPORTANT: they are not "obvious" statements and not statements that "do not require the proof". We just have to accept them to build a hierarchy of logical conclusions upon them because we have to have some statements in the foundation.
Full description of axiomatic approach to Geometry and proofs of angle-side-angle (ASA) and side-side-side (SSS) theorems about congruence of triangles can be found at Unizor by following the menu items Geometry - Triangles - Congruence.