Indeed, if you look at the behavior of a Bloch wave function and how it behaves for a wave vector at the Brillouin zone boundary you will find that you can construct a standing wave as the result of superposition of two wave with wave vector of pointing in opposite direction. This is the same way of creating  a classical standing wave as superposition of two traveling waves with proper phase shift  propagating in opposing directions. In the case of the electrons with wave vector at the zone boundary you can create two types of standing waves. One where there is a lot of electron density between the positive ions and one where the electron density is peaking at the position of the positive ions. These two different standing waves have obviously different energies (like bounding and antibonding in the case of molecular orbitals). It is this energy difference that creates the band gap. The band gap obviously depends on the fact that the potential the electrons see is periodic with a certain corrugation (there is no band gap for free electrons in a constant potential a periodic variation of the potential creates the band gap). Therefore the magnitude of the band gap is determined in first approximation by the leading Fourier coefficient of the periodic potential. 

Realy the link  which you appointed to is very interesting

(https://www.researchgate.net/go.Deref.html?url=http%3A%2F%2Fnptel.ac.in%2Fcourses%2F113106040%2FLecture32.pdf) 

If you read it carfully you can find the answer  easily without hesitate.

There is a relation between the wave propagation k and the wavelength so there is relation between E and k over m for free electron and in the Brillouin zone boundary as appointed in the paper

I have always hated these sorts of explanations and I understand why you find it confusing.  The actual band gaps are not well approximated by nearly free electron theory but by Green's function methods.  This implies the right way to think about what is going on here is that the states near the BZ are mixing more strongly in a correlated way so that there is a split.  What does this have to do with Bragg diffraction?  The edges of the zones are where Bragg diffraction occurs.  Does this mean that this diffraction "causes" the splitting?  I find this suggestion kind of a stretch.  What matters is that this is exactly where the interactions of the states is the largest so they can mix.

The usual nearly free electron model makes more sense for neutral fermions in periodic optical potentials for ultracold gases.  At least here the necessity of treating electrons near the fermi surface as weakly interacting quasiparticles is not necessary.  Interestingly, we usually apply the nearly free electron theory to explain gaps that are not so near the fermi surface anyway where we don't expect the quasiparticle assumption to be valid. 

Maybe you will like the optical analog better.  When light passes through a dielectric what "causes" the band gaps to open up?  One can view the diverging internal energy storage of the the electromagnetic wave near resonance as the driver of such forbidden frequencies in the spectrum.  Such frequencies can only generate evanescent waves.  I made a pretty picture of this in the paper below.  

Data Abraham Minkowskii 4