Start by asking the learners to consider the question "How far forward would a bicycle travel if its wheels turned through one complete revolution?" Then show the first part of the problem, and ask what the difference is between the bicycle problem and the coin in the box problem - this should make it clear that the corners are key.
Now show the second diagram. Intuition may suggest that if the coin is travelling on two sides, each side would not need to be as long in order to get the whole circumference to touch, but having a corner where part of the circumference doesn't touch makes things interesting! Learners could draw corners on paper and roll a cardboard circle along them, highlighting on their circle the parts
that touch and the parts that don't.
Now look at what happens when the coin rolls around the inside of a tray. Ask the learners to discuss in pairs whether all of the circumference of the coin will touch on one circuit of a $4$ by $3$ tray, and then share ideas.
Try the problem Five Circuits, Seven Spins.
There are more ideas, explanations and problems to work on in the article A Rolling Disc.
How many times does the disc rotate about its own centre when it makes one revolution around the tray?
What happens if corners are not $90^\circ$?
Physically manipulating a circular object inside a frame can make it clearer what's happening at the corners.
The problem Rolling Around may be a good starting point.