Example 3 As shown in Figure 3, given that the volume of the regular quadrilateral pyramid $P$ $A B C D$ is 1, $E, F, G, H$ are the midpoints of line segments $A B, C D, P B, P C$ respectively. Then the volume of the polyhedron $B E G-C F H$ is . $\qquad$
Take the midpoints $M, N$ of $BC, EF$ respectively, and connect $MN$, $GM$, $GN$.

Then the polyhedron $BEG-CFH$ is divided into a quadrilateral pyramid $G-EBMN$ and a triangular prism $HFC-GNM$.

Since $E, F, G, H$ are the midpoints of segments $AB, CD, PB, PC$ respectively, and the volume of the regular quadrilateral pyramid $P-ABCD$ is 1, therefore, $=\frac{1}{8}+\frac{3}{16}=\frac{5}{16}$.