The spin statistic theorem said that there is a relation of this label (the spin) and the statistic of the particle (if is a fermion or a boson) and that relation is that half integer spin (1/2,3/2,5/2...) particles are fermions and integer spin particles are bosons (0,1,2,3...). The importance is that by measuring the spin of the particle we now know if the particle is a boson or a fermion. For example the proton, the electron and the neutron have spin one half then they are fermions. Is really amazing. By knowing the spin of the particles we know how it will behaves.
How can you proof the theorem? Well the usual proof requires Quantum Field Theory. You need to assume that the theory is Lorentz Invariant, which is a elegant way to say that it agrees with the Einstein Theory of special relativity. Also you assume that the theory is microcasual. That means that two things that are at different point at the same time cannot interact.
The things that you need to know is that a quantum field theory for bosons and for fermions are different, because they behave differently. Formally we said that we need to use commutations relations to create the fock space of the bosons and anti-commutation relations to create the fock space of the fermions. Also is different to created a theory for each spin label. For example for spin-one half we need to use spinors field, for spin zero we need to use scalars field and spin one we need to use vector fields, ect. Then when we construct a theory for fermions with spin one-half we need to use spinors and anit-commutation relations. That means that by imposing spin and statistics (fermion or boson) at the same-time we constraint the theory a lot and this is the trick of the proof.
The proof of the theorem is to show that a theory that have a boson with spin half integer will be inconsistent (will have problems and should be rejected) and the same for a theory with fermions with integer spin. Then the only option is that the bosons have integer spin (0,1,2,3..) and the fermions half integer spin (1/2,3/2,5/2...).
This is a formal result of the theory, but the best thing is that all experimental results agree. All the observed fermions have half integer spin and all the bosons have integer spin. Thousand and thousand of experiments verified that this result is correct.
I hope it helps.