Interpretation of the vector current in field theory

Well, such form follows from the Noether theorem, which states that if action (in Minkowski space-time) $$ S = int d^{4}xL $$ of fermions $Psi$ is invariant under transformation $$ tag 1 Psi to e^{ialpha}Psi , quad bar{Psi} to e^{-ialpha}bar{Psi}, $$ then exists the object $J$ with one vector indice, namely $$ tag 2 J_{mu} = ifrac{partial L}{partial (partial^{mu}Psi)}alpha Psi - ialpha bar{Psi}frac{partial L}{partial (partial^{mu}bar{Psi})}, $$ which is conserved: $$ partial_{mu}J^{mu} = 0 $$ Conservation law can be rewritten in the form $$ frac{d}{dt}int_{Omega} d^{3}mathbf r J^{0} equiv frac{dQ}{dt} = -int_{dOmega} dmathbf {S}cdot mathbf J, $$ which means that the changing with time the quantity $int J^{0}d^{3}mathbf r$ in the given volume $Omega$ is related to the flux of quantity $mathbf J$ through surface $dOmega$.

The scalar quantity $Q$ is thus called charge (with $J^{0}$ being charge density), while the quantity $mathbf J$ is current density.

The vector nature of $J^{mu}$ follows from the fact that the transformation $(1)$ isn't related to space-time transfrormations, while indices are. So the only quantity which can be $J$ in our case is the vector (one indice comes from expilcit expression for Noether current).

As for explicit form of current $J^{mu}$, let's use particular lagrangian of free fermions, $$ L = bar{Psi}(igamma^{mu}partial_{mu} - m)Psi , $$ with $gamma_{mu}$ being Dirac matrices, $Psi$ being Dirac spinor, $bar{Psi} equiv Psi^{dagger}gamma^{0}$ being Dirac conjugated spinor. Such lagrangian is constructed from the requirement of Lorentz invariance and from requirement that we obtain Dirac equation for field with spin $frac{1}{2}$ after using Euler-Lagrange equations.

and insert it into $(2)$. You'll immediately obtain the result (up to the sign, which isn't relevant here).