Fit the nine pieces into the frame, stacking them up tidily to form a regular tetrahedron, as shown.
This may be difficult, particularly if you don't have a physical set of the pieces.
However, with the right approach, it's easy – maybe easy enough that I can convince you of the solution without use of any physical pieces.
First, we observe that the 5-long pieces can only be placed along edges of the tetrahedron. Then observe that as they can't share a vertex, there's only one way they can both be accommodated in the tetrahedron. Put them there.
Now we will deal with the 4-long pieces. Therre's only four places the now partly-filled tetrahedron can accommodate them. So put them there.
Now we find that we're left with a 3×3 square and three 3-long pieces to fill it. (If we're trying to visualise the structure this is difficult.)
The diagam to the right assumes that we have assembled the tetrahedron, and are holding it so that the two 5-long pieces are both horizontal. It shows the five horizontal "slices" that now make up the tetrahedron.
This answers question 2 at A technique in solving puzzles.