Solution:

We can clearly cut the polygon into $12$ strips width $1$, so the smallest number is $\leq 12$.

There are $84$ unit squares in the figure. Each cut along the edge of a unit square not already cut (and not on the boundary) increases the number of pieces by at most $1$. So it is sufficient to show that at most $72$ edges remain uncut (after cutting into polygons). Because then cutting the remaining edges would increase the total number of pieces by at most $72$. But the final number of pieces is $84$, so we would have to start with at least $12$.

Initially, there are $144$ edges, so we have to show that at least $72$ of them are cut to make the polygons. An interior vertex has $4$ edges. At least two of them must be cut, or the vertex would be the center of an uncut $2 \times 2$ square. If we take alternate interior vertices ($36$ in total, as shown below), then each has at least two cut edges, so in total at least $72$ edges are cut to make the polygons.