How does quantum mechanics and quantum field theory explains discrete energy levels of particles? Please give me a brief explanation as to how qm and qft describe and explain the energy level that exist in an atom.
I understand that in QM energy state is quantized but does not offer an explanation as to why, does qft solves this phenomenon? 
 A: Feynman two pennies worth on why questions.

Why are the energy levels in atoms quantised? Because the universe is
  made as such.

A complete different point is how such description is achieved in quantum mechanics. The energy levels of a quantum system are defined as the spectrum of its hamiltonian: an entire area of mathematics deals with the relations between operators, their eigenstates and their spectra. Quantised energy levels, i. e. discrete spectrum, is often connected to bound eigenstates of the operator, therefore we interpret this aspect as bound physical states of atoms having discrete energy levels, though it needs not necessarily be so. In fact, scattering states (often unbounded) do present continuum spectrum, as many other physical systems do.
A: 
how does quantum mechanics ... explain discrete energy levels of particles?

If you solve the Schrodinger equation, which is a law that can not be demonstrated, for the hydrogen atom or for any atom or molecule you will obtain discrete solutions, discrete energies. That is all.
If we admit that the Schrodinger equation governs the world of atomic particles, and all experiments demonstrate it does, the rest is just math and those discrete energies result simply by solving the equation.
