$L\in NC^1$ iff there exists a sequence of poly sized formulas that decides $L$
Let the size of a circuit or formula be the number of leaves. The total number of vertices in a formula is at most twice its size.

You can prove by induction the following claim:


  If a function is computed by a fan-in 2 circuit of size $S$ and depth $D$, then it is computed by a formula of size $2^D S$.


If $L$ is in NC1 then its $n$-bit fragments have circuits of size $n^{O(1)}$ and depth $O(\log n)$. The claim shows that there are equivalent formulas of size $n^{O(1)} 2^{O(\log n)} = n^{O(1)}$.