Recently I read, that only connected Feynman diagrams give contribution of nonzero values into the scattering amplitude. Why it is so and what is the physical sense of connected diagrams (due to their definition in Wikipedia)?
Also, I don't understand why strongly connected (=one-particle irreducible) Feynman diagrams are so important in scattering theory. By the other words, I don't understand why do we cut off one of the internal lines in the Diagram and does it relate to some physical process.
The fact that only connected Feynman diagrams contribute to the scattering amplitude can be interpreted in terms of the vacuum of the theory. Omitting disconnected diagrams amounts to shifting the vacuum: the vacuum of the interacting theory differs from that of the free theory.
Regarding your second question: strongly connected (also called one-particle irreducible) diagrams are needed in order to calculate loop corrections to the propagator. The exact propagator is given by a geometric series consisting of one-particle irreducible diagrams. Furthermore, they play a role in the calculation of the exact vertex function.