Incorporate spinor in field equation I got problem understanding the concept as it state
1. Spinors do not work with the principle of General covariance. But how and why?
2. Contracting spinor into the tetrad solves this delemma.
 Anybody explaining this concept would be helpful.
 A: The short answer: spinors transform nontrivially under local Lorentz boosts/rotations.

Spinors do not work with the principle of General covariance. 

More precisely, it's the local Lorentz covariance that is the focus point here. 

Contracting spinor into the tetrad solves this delemma.

In addition to tetrad $e$, we also need spin connection $\omega$ to write the Lorentz-covariant derivative $D\psi = (d + \omega)\psi$ (usually spin connection $\omega$ is expressed as a function of tetrad $e$ via the zero torsion condition $T = de + e\wedge w + w\wedge e = 0$, which does not fly here since spin currents introduce non zero torsion).
The reality is that tetrad $e = e^I_\mu\gamma_Idx^\mu$ is always there even for nonspinors, only that it's hidden from the plain sight via contracting into the metric:
$$
g_{\mu\nu} = e^I_\mu e^J_\nu \eta_{IJ}.
$$
The thing with spinor action
$$
S_{spinor} \sim  \int{i\bar{\psi}e\wedge e \wedge e \wedge (d+\omega)\psi}
$$
is that there are 3 tetrads (or 1 co-tetrad, if you will) and there is no way contracting odd number of tetrads into metric.
