The propagator in the language of path integral quantization is the inverse of whatever operator is in the quadratic part of the action \(S\). For example, for the Klein-Gordon field, the action (up to a total derivative) $$ S = -\frac{1}{2} \int \phi\left(\partial_\mu\partial^\mu+m^2\right)\phi. $$ Thus the propagator is $$ \Delta_F(x,y) = \left(\partial_\mu\partial^\mu+m^2\right)^{-1}. $$ In momentum space $$ \Delta_F(k) = \frac{1}{k^2-m^2+i\epsilon}.$$

The Dirac action $$ S_D = \int \bar{\psi} (i\gamma^\mu\partial_\mu-m) \psi. $$ In Dirac/Feynman slash notation, contraction with the \(\gamma\) matrices will turn into a slash over the vector being contracted. The fermion propagator in momentum space is $$ S_F(k) =  (\gamma^\mu k_\mu-m+i\epsilon)^{-1} = \det{ (\gamma^\mu \partial_\mu-m+i\epsilon)}=\frac{\gamma^\mu k_\mu+m}{k^2-m^2+i\epsilon}.$$

Notice $$ -(\gamma^\mu \partial_\mu-m) (\gamma^\mu \partial_\mu+m) = \partial_\mu\partial^\mu+m^2. $$ That is why Dirac said he was “trying to take a square root” while inventing his equation.