As noted, it is actually the foam that is the spring. I don't think it is clear exactly what the plate does, other
than it does seem to be needed.
As above, getting the spring part right is actually pretty complicated. Trying another way to explain it.
You foot hits the ground with what, like 4x your weight or something, and that is when the spring will
compress. As soon as that force lessons, which happens well before you are pushing off, the 'spring'
will start decompressing. To take the extremes, if it fully decompresses well before you start your
push off you won't get any energy back. If it decompresses after your foot leaves the ground you
won't get any energy back either.
One explanation that I've heard for why most elite runners have about a 180 cadence is that that
matches what is needed to make best use of the main natural spring used in running, the Achilles tendon.