In this paper, we identify the scaling limit of the fermionic discrete Gaussian free field (fDGFF) as a logarithmic conformal field theory (CFT) in two dimensions. We first establish a one-to-one correspondence between the space of local observables of the fDGFF and the space of local fields of the symplectic fermions CFT, a logarithmic CFT with central charge $c = - 2$. This correspondence is meaningful in the sense that, when appropriately renormalised, the fDGFF correlation functions converge to corresponding CFT correlation functions in the scaling limit. As an application to these results, we interpret (the scaling limit of) certain local observables in the uniform spanning tree and the Abelian sandpile model as local fields of the symplectic fermions.