How does Stern-Gerlach experiment lead to conclusion of electron's spin?

The Stern-Gerlach experiment is about angular momentum in general, not particularly about the electron spin.

The Stern–Gerlach experiment demonstrated that the spatial orientation of angular momentum is quantized. Thus an atomic-scale system was shown to have intrinsically quantum properties. In the original experiment, silver atoms were sent through a spatially varying magnetic field, which deflected them before they struck a detector screen, such as a glass slide. Particles with non-zero magnetic moment are deflected, due to the magnetic field gradient, from a straight path. The screen reveals discrete points of accumulation, rather than a continuous distribution,

Even though the result of the Stern−Gerlach experiment has later turned out to be in agreement with the predictions of quantum mechanics for a spin-​1⁄2 particle, the experiment should be seen as a corroboration of the Bohr–Sommerfeld theory.

Italics mine for emphasis.

Note: turned out to be in agreement., this, when not stressed, is what creates the confusion with the SG.

So actually the Stern Gerlach shows quantization of angular momentum, the conclusion about the electron's spin needs the Bohr model, and the other definitive experiment of the fine structure of the hydrogen spectrum lines, as discussed here:

Two types of experimental evidence which arose in the 1920s suggested an additional property of the electron. One was the closely spaced splitting of the hydrogen spectral lines, called fine structure. The other was the Stern-Gerlach experiment which showed in 1922 that a beam of silver atoms directed through an inhomogeneous magnetic field would be forced into two beams. Both of these experimental situations were consistent with the possession of an intrinsic angular momentum and a magnetic >moment by individual electrons. Classically this could occur if the electron were a spinning ball of charge, and this property was called electron spin.

Quantization of angular momentum had already arisen for orbital angular momentum, and if this electron spin behaved the same way, an angular momentum quantum number s = 1/2 was required to give just two states.

Lets see your questions:

And I couldn't understand how does that concludes the spin of electrons

as quoted above, the conclusion about the spin of the electron needed the evidence of the fine structure of the atomic spectra. It is sloppy to talk about a conclusion just by the SG experiment, but unfortunately it happens. SG establishes the quantization needed for the Bohr model, electron spin needs further evidence

because there may be an effect of proton's spin

The protons are bound in the silver atom , and the nucleus is shielded by the electron orbits so it cannot be that.It is the neutral silver atom's as a whole that the dipole moment is checked with the experiment.

or maybe silver atoms may interact with each other and act as dipoles (if the atom is not symmetric then how they can eliminate this possibility)?

It is a beam of neutral silver atoms going through an inhomogeneous magnetic field, see the description here. In a beam the atoms are separate, non interacting.

I hope this helps.

Edit. Answering the other question about electron spin, I found this arxiv paper , which shows data for a free electron SG experiment that shows two peaks when the magnetic field is turned on.