Mathematical significance of the "Dirac conjugate"

The Lorentz group, $SO(1,3)$, is non-compact, thus their representations are not unitary (in general).

Therefore, if you have a spinor, $\psi\in \mathcal{S}$, transforming as $\psi\mapsto S\psi$, it follows that the construction $$ \psi^\dagger \psi \mapsto \psi^\dagger S^\dagger S \psi \neq\psi^\dagger \psi,$$ since $S^\dagger\neq S^{-1}$.

This tells us that $\psi^\dagger$ does not belongs to the dual space of the spinors, $\psi^\dagger\not\in \mathcal{S}^*$.

It this point, you can realize that $$S^\dagger\gamma^0 = \gamma^0 S^{-1},$$ and this allows to define a dual spinor to $\psi$ through the construction $$\mathcal{S}^*\ni\bar{\psi}\equiv \psi^\dagger\gamma^0.$$

Hope it would be helpful.

Summary

The Dirac conjugate serves to define a dual spinor, by giving a spinor in the direct space.