How did Paul Dirac predict the existence of antiproton?

The Dirac equation implies negative energies as well as positive. This is due to energy-momentum relation $E=\pm \sqrt{m^2+p^2 }$. If we replace $E$ and $p$ by operators $E\to i\frac{\partial }{\partial t}$ and $p\to -i\nabla$ we get the Klein-Gordon equation $(\Box+m^2)\phi=0$ for scalar (spinless) fields $\phi$. The problem with this equation is that it gives solutions with negative probability density and negative energy.

In order to overpass the problem with negative probability density, Dirac made the K-G equation linear in time derivative $\frac{\partial }{\partial t}$ (and in space derivative to make it covariant). So he get the following equation (the Dirac equation): $(i\gamma^{\mu}\partial_{\mu}-m)\psi=0$ for particles with 1/2-spin (fermions: electrons, protons etc.).

The Dirac equation gives positive probability densities which is good, but the problem with negative energy quantum states remained. To overpass this problem, Dirac postulated that the universe is filled with infinitely dense "sea" of negative energy particles (electrons), the Dirac sea. Due to Pauli exclusion principle no other electron can fall into the Dirac sea, but sometimes one electron can leave the Dirac sea creating a hole which would act like positive energy electron with opposite charge - the positron, experimentally discovered by Carl Anderson. This holes are called antiparticles.

But the Dirac sea theory has some problems, like the problem of infinite charge of the universe and the fact that the bosons, which have antiparticles too, do not obey the Pauli exclusion principle and the hole theory doesn't work for them. This problems are solved in quantum field theory.

You'll find Dirac's 1933 Nobel lecture on the Nobelprize.org website. The pdf is quite brief (5 pages long) and speaks on the antiproton at the end (p4). The argument is the following :

In any case I think it is probable that negative protons can exist, since as far as the theory is yet definite, there is a complete and perfect symmetry between positive and negative electric charge, and if this symmetry is really fundamental in nature, it must be possible to reverse the charge on any kind of particle.

In short, the particle-antiparticle symmetry seems to be a law of nature, so the proton should also have an an anti-proton partner.

Edited to expand below

@ANKU's answer above is the answer to the question "How did Dirac predict the existence of antimatter", and this work was done for the electron (or the positron). Once the positron has been predicted and and observed, came the intuition that this symmetry was much more general. However, in 1933, Dirac didn't think that this theory could be directly applied to protons. To quote his Nobel lecture :

The theory of electrons and positrons which I have just outlined is a self- consistent theory which fits the experimental facts so far as is yet known. One would like to have an equally satisfactory theory for protons. One might perhaps think that the same theory could be applied to protons. This would require the possibility of existence of negatively charged protons forming a mirror-image of the usual positively charged ones. There is, however, some recent experimental evidence obtained by Stern about the spin magnetic moment of the proton, which conflicts with this theory for the proton. As the proton is so much heavier than the electron, it is quite likely that it requires some more complicated theory, though one cannot at the present time say what this theory is.