Answer: 750

Small Hint:

The four triangular points fold up along the sides of $S',$ and their tips meet at the apex above the center of $S'$

Big Hint:

The slant from the midpoint of a side of $S'$ to the star tip has length $20 - \frac{15}{2};$ use a right triangle through the center to get the height

Solution:

Folding the star along the sides of $S'$ lifts the four triangular points so that their tips (the midpoints of the sides of $S$) meet at a single apex $V.$ Let $M$ be the center of $S'$ and $P$ the midpoint of one of its sides. In the flat figure, the distance from $P$ to the tip of its triangle is $20 - \frac{15}{2} = \frac{25}{2},$ and this becomes the slant $PV$ after folding.

Triangle $PMV$ has a right angle at $M,$ with $PM = \frac{15}{2},$ so the height is $$\begin{aligned} VM &= \sqrt{\left(\tfrac{25}{2}\right)^2 - \left(\tfrac{15}{2}\right)^2} \\ &= \sqrt{100} = 10. \end{aligned}$$ The volume is $\frac{1}{3} \cdot 15^2 \cdot 10 = 750.$