There is some little historical reference.
The Dirac equation has two linearly solutions which are the eigenstates of Dirac hamiltonian: first refers to the positive values of energy while the second refers to the negative ones. In classical field theory we can violate this problem by setting the trivial initial conditions $\psi (\mathbf r , t) $ for negative values, but in quantum theory it doesn't predict the absense of "negative" states, because there is possible discrete transitions between "positive" and "negative" states. The problem of negative energies is very important, because the absense of restrictions on the negative energy states promotes things like perpetuum mobile. It is experimentally unobserved.
Dirac phenomenologically solved this problem by adding the Dirac sea conceprion to the free field theory. This gives the infinite number of bonded particles with negative energies, which makes possible the absence of free solutions with negative energies due to Pauli principle. But simultaneously it determines the processes of annihilating or processing particles. For example, if there is the idle state in the vacuum with negative energy $-E$, the free electron with energy $2E$ can take this state with emission of energy $2E$ (it is equal to annihilation of particles). So there is possible of decreasing of total number and charge of free particles.
But there's nothing critical for interpreting the Dirac theory as one-particle, because the bonded electrons can't interact between each other: interaction means transition from one bonded state to another one, but this is impossible because of the Pauli principle. So in the case of free Dirac particles this infinite number is unobservable due to isotropy and homogeneity of the space-time, and even after assuming the Dirac sea conception you can still interpret the Dirac equation as single-particle one. But lets introduce some external field. It can interact with the Dirac sea and pull out the particles from it. So the Dirac theory stops being one-particle one when the external field may give ene energy more than $2mc^2$.
All of these thinking leaded to the quantum field theory. All the ideas and problems which I described are still present in quantization formalizm (infinite vacuum energy, creation and destruction operators, scattering processes with pairs creation in an external field etc.), but it is important to distinguish the reasons and consequences. In our world there aren't free particles with negative energies - this is a reason why we introduce the quantum field theory.