Suppose you have a rational map $f$ from a smooth projective surface $S$ to $\mathbb{P}^{n}$ given by some linear system $V \subset |D|$. Without loss of generality you may assume that the image of $f$ is non-degenerate, i.e. is not contained in any hyperplane. In that case we can subtract the fixed component of $V$ to make the base locus of $V$ is at most finite. Now take a base point of $V$ and blow it up $g : \widehat{S} \to S$. The pullback $g^{*}V$ will consist of divisors in $|g^{*}D|$ which contain a positive multiple $nE$ of the exceptional divisor $E$. So $g^{*}V$ has a fixed component and therefore is of the form $\widehat{V} + nE$, where $\widehat{V} \subset |g^{*}D - nE|$. The linear system $\widehat{V}$ gives a rational map $\hat{f}$ from $\widehat{S}$ to $\mathbb{P}^{n}$ which has no fixed component and coincides with the original $f$ on $\widehat{S} - E$. Now set $\widehat{D} = g^{*}D - nE$. Note that the fact that $\widehat{D}$ has no fixed component implies that $\widehat{D}^{2} = D^{2} - n^{2} < D^{2}$. If $\hat{f}$ has no base point, then we are done. If it does have base points, then we can repeat the process. Since the degree of the linear system decreases on each step the process will terminate.
Suppose you have a rational map $f$ from a smooth projective surface $S$ to $\mathbb{P}^{n}$ given by some linear system $V \subset |D|$. Without loss of generality you may assume that the image of $f$ is non-degenerate, i.e. is not contained in any hyperplane. In that case the base locus of $V$ is at most finite. Now take a base point of $V$ and blow it up $g : \widehat{S} \to S$. The pullback $g^{*}V$ will consist of divisors in $|g^{*}D|$ which contain a positive multiple $nE$ of the exceptional divisor $E$. So $g^{*}V$ has a fixed component and therefore is of the form $\widehat{V} + nE$, where $\widehat{V} \subset |g^{*}D - nE|$. The linear system $\widehat{V}$ gives a rational map $\hat{f}$ from $\widehat{S}$ to $\mathbb{P}^{n}$ which has no fixed component and coincides with the original $f$ on $\widehat{S} - E$. Now set $\widehat{D} = g^{*}D - nE$. Note that the fact that $\widehat{D}$ has no fixed component implies that $\widehat{D}^{2} = D^{2} - n^{2} < D^{2}$. If $\hat{f}$ has no base point, then we are done. If it does have base points, then we can repeat the process. Since the degree of the linear system decreases on each step the process will terminate.