What is the mechanism of formation of Grignard's Reagent?
Looking at the mechanism:



We see that these are not double headed arrows, but single headed arrows. This indicates the movement of single electrons, rather than electron pairs as is common in most mechanisms.

Mechanisms that involve movements of single electrons tend to form free-radicals.

The lewis structure of Mg has the electrons seperated in order to make it clearer to read.

In step 1, the shared electron pair in $\ce{R-X}$ has been split to form two radicals: $\ce{R}{\small{•}}$ and $\ce{X}{\small{•}}$. However the $\ce{X}{\small{•}}$ radical is further attacked by an electron, this time coming from the $\ce{Mg}$. Thus forming a free-radical magnesium halide: $\ce{X^-Mg^+}{\small{•}}$.

After this, the two free-radicals undergo a termination step to form the Grignard reagent.