$X_i equiv a_i pmod{P}$ for some $a_i in mathcal{O}$ given a prime ideal $P$ of $mathcal{O}[X_1, ldots, X_n]/(f_1, ..., f_n)$

This is the multivariate form of Hensel's Lemma. The isomorphism $R_mathfrak{P}/mathfrak{P}R_mathfrak{P} to mathcal{O}/mathfrak{m}$ is the same as a choice of $n$ elements $overline{a_i} in mathcal{O}/mathfrak{m}$ (they're just the images of $X_i$) and what you want is a solution of the polynomial system of equations $forall i: f_i(X_1,ldots,X_n) - f_i(a_1,ldots,a_n) = 0$ to which you already know an approximate solution, namely $(overline{a_1},ldots,overline{a_n})$. The condition for the determinant is exactly the non-degeneracy condition for the approximate solution that you need to apply Hensel lifting.