Why outer measure?

If you approximate from outside you get an outer measure that is subadditive: $\mu^\ast[A \cup B] \le \mu^\ast[A] + \mu^\ast[B]$. This allows to think of measure as some kind of a "norm" of a set, and, in particular, define a metric $\rho(A,B) := \mu^\ast[A \vartriangle B]$, where $\vartriangle$ is symmetric difference. Completion and extension by continuity using this metric gives you measure theory (by this I mean that, first of all, the measure algebra is just the completion of the space of "simple" sets by this metric, and measurable sets are exactly those that can be approximated by "simple" ones using this metric).

Now if you try to approximate from inside, you get an inequality in the wrong direction: for $A$ and $B$ disjoint $\mu[A \cup B] \ge \mu[A] + \mu[B]$. So there is no metric, no completion, ..., no measure theory. In particular, this doesn't give any straightforward way to select which sets are measurable and map them to elements of the measure algebra.