What are the parameters of protonation of a molecule? Adding an acid replaces the $\ce{-MgBr+}$ with an $\ce{H+}$. Why does this happen? Is $\ce{H+}$ a better electrophile than $\ce{MgBr+}$?

Or, to put it another way, why doesn't $\ce{MgBr}$ stays attached to $\ce{O}$?
We are taking about electrophilicity of ions.
It basically depends upon:
1. charge on the ion(+)
2. density of plus charge on it  

So comparing $\ce{H+}$ and $\ce{CH3+}$, the former has a higher plus charge density so that's why it will be more attracted towards 'O'.