Q:

How were Electromagnetism and Weak interaction unified and proven?

How did the electroweak interaction come into fruition?

A:

I'm going to assume that you're nontechnical, and sketch a generalized story of why people would think that the weak interaction would have something to do with electromagnetism.  This is going to be a little metaphorical, and very non-technical, and I'm likely to get some things wrong.  Please comment as such, and I will fix it.  Also note that I am necessarily going to leave important details out of this discussion.  This is not a primer on electroweak theory, this is a motivation for understanding the original question.
So, the first thing was Fermi's prediction of the neutrino.  This came about because careful measurements showed that the decay of a neutron to a proton and electron appeared to not conserve momentum.  Because the observed decay appeared to conserve charge, this required that the decay product have no charge.  Conservation principles also indicated a particle with very little mass.  Fermi therefore proposed the name "neutrino" for a very light particle that was a product of this decay and that carried away the missing momentum.  He proposed an interaction where four point particles simultaneously interacted, and this seemed to work relatively well, and the notion of the "weak nuclear force" was born, since this interaction was apparently short-ranged.
Time passed, and perturbation theory became more sophisticated.  It was discovered that Fermi's theory of the weak interaction was self-inconsistent at higher energies than the neutron decay.  Furthermore, symmetries were discovered between certain pairs of particles -- for example, at energies much higher than the electron mass, the weak force appeared to treat electrons and neutrinos in a way that they were approximately interchangeable.  Similar effects happened for several other pairs of particles.  From group theory, we know that this type of symmetry is described by the group $SU(2)$, and we understood the dynamics of the electromagnetic $U(1)$ theory pretty well by the '50s, so it seemed logical to try to write down a $SU(2)$ theory of the weak force, and get rid of the four-fermion interaction with a weak-force equivalent of the photon (which woudl actually be three bosons, rather than one).
This, however, had two problems:
1) there was no way to write down a consistent gauge theory that was short-ranged -- this required giving the bosons mass, and the mass broke the gauge magic
2) Several decay channels were discovered that indicated that the gauge bosons had to come in both charged and uncharged varieties, in order to support processes like:
$$e^{-} + {\bar \nu_{e}} \rightarrow \mu^{-} + {\bar \nu_{\mu}}$$
as well as ones like the weak correction to:
$$\mu^{-} + \mu^{+} \rightarrow e^{+} + e^{-}$$
But it was unclear how to make the gauge bosons come in both electrically charged and uncharged varieties.
So, both of these problems were solved with the introduction of the unification of the electromagnetic force with the weak force.  The short story is that a group large enough to contain both was created.  Then, the Higgs boson was added, and the key thing here is that the theory was constructed in such a way the Higgs boson had a nonzero value in vacuum, and this picked out a "special" direction in the $SU(2) x U(1)$ space that forced mixing, and left one massless boson, and three massive ones, while also leaving two uncharged bosons (one of which was the massless one), and two charged ones.  Because all of these properties were emergent from the Higgs symmetry breaking, and not fundamental, it also solved the problems with inconsistency and the massive bosons breaking gauge -- the mass was just an apparent thing created by the Higgs.  
The other thing worth nothing is that the $U(1)$ in electromagnetism is a "left-over" massless degree of freedom that is in a mixed direction within the $SU(2) x U(1)$, it is NOT the $U(1)$ in the electroweak symmetry.  This mixing gives is what gives the weak bosons their electric charge.
I hope this was helpful.  I tried to walk the line between handwaving too much and being too overtechnical.

