Wrap the edges of fabric around the foam core, pulling the material taut and securing it with painter's or masking tape.
Place the wrapped sheet of foam core in the frame, so the colorful side shows through the front opening. Secure the foam core by re-attaching the back panel of the frame with the spring clips or turning sliders, whatever came with the frame.
The foam core is thick, so if the turn button/slider gizmos don't want to slide into place, you may need to trim the foam core a little more to provide enough of a gap. Just be careful not to trim the foam core so much that it pops right through the frame. (Ahem. Duct tape works really well for this too and it won't show from the front. It can be our little secret.)