This was solved by Ling Xiang Ning, Allan from Raffles Institution, Singapore and Tony and John from State College Area High School, Pennsylvania, USA.

This is Allan's solution: "The first part of the animation shows that a square of size $a^2$ subtracted off a square of size $b^2$ would result in a rectangle of sides $(b - a)(a + b - a + a)= (b - a)(b + a)$. We can see that the difference of the two squares $b^2 - a^2$ is the same area as the area of a rectangle $(b - a)(b + a)$.

Therefore $b^2 - a^2 = (b-a) (b+a)$."

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