The fact that a definite value is found in every measurement of a physical observable does not simply follow from the formalism of quantum mechanics; it has to be required separately as an additional postulate.
The measurement postulate states that a single one of the possible measured values is found. But what are the possible measured values? They are the eigenvalues of the measured observable.
The concept of the eigenvalue is treated in more detail later (Lesson 8.6, Lesson 9.2). The measurement postulate is the reason why eigenvalues and the solving of eigenvalue equations play such a major role in quantum mechanics: they are the possible measured values.