How does de Broglie actually prove Bohr's postulates?
de Brogile explains why orbitals are quantised

Strictly speaking de Brogile doesn't prove Bohr's postulates which are mostly wrong. But he did provide an explanation for the most important of Bohr's ideas: electron orbitals are quantised.

Bohr's whole model starts with the classical idea that electrons "orbit" a nucleus. But this has several problems not least of which is why don't they just radiate away their energy and fall into the nucleus (which is what accelerating charges should do in classical electromagnetism). Bohr made the extra assumption that the electrons around a nucleus could only "orbit" in a small number of fixed paths or orbitals which explained the actual observations. He didn't have any good reason why this was true.

de Brogile provided a reason. According to his postulates things like electrons are not just classical particles but have some of the characteristics of waves. and the thing about waves is that they interfere with each other. Electrons orbiting a nucleus can only do so if their wave-like property fits into a specific orbit (the way the vibrations in a guitar string produce only specific notes made from standing waves of integer divisions of the string length). Imagine integer numbers of electron waves around a nucleus and you are a third of the way of getting the picture (electrons orbit in 3D so the compatible vibrations are spherical harmonics in 3D space not the easier to picture vibrations of a 1D guitar string).

de Brogile, in short, produced a justification for the Bohr assumption that electron orbits had to be quantised.

Those pictures you see of electron "orbitals" are, in essence, pictures of the possible "waves" electrons can form when orbiting a nucleus. 

All the above simplifies things a bit but the key ideas were from de Brogile and were more fully mathematicised by Dirac and Schroedinger giving us the modern quantum picture of how electrons behave.