How do we know that the fundamental forces become unified as the energy of particle interactions increases?

The standard model of particle physics defines the interactions of fundamental forces as being due to particles called bosons.

The electromagnetic force is propagated by the photon, which is a boson.

The weak nuclear force has two bosons. The W is a charged particle which is responsible for changing a quark from once flavour to another. This is what causes beta decay where a neutron is turned into proton, en electron and an electron antineutrino. The other boson is the Z which is involved when neutrinos are scattered off other particles.

Both forces are described mathematically by symmetry groups. When the W and Z were discovered it was possible to prove that electromagnetism and the weak force can be unified into the electroweak theory. It worked by creating a unifying symmetry group which shows that the photon and the Z boson are indistinguishable at high energies.

The next step is to discover a Grand Unified Theory (GUT) which unifies the colour force which binds quarks together with the electroweak force. The strong nuclear force is actually a residual effect of the colour force.

There are several candidate theories for a GUT. The problem is that the unification energy if so high that we can't conceive of how to build a particle accelerator to reach it. Candidate GUTs require additional particles which we haven't yet discovered.

It is likely that a GUT will be found but not soon.

Gravity poses a real problem. There is no quantum theory of gravity. A Theory of Everything which unifies all four forces requires new physics. String theory made be the answer but it has a long way to go.