It follows from Cavalieri's Principle, or else if you know that shear transforms have determinant $1$, and hence don't change area, that's another way to see it.
I don't understand why it isn't "easy to see" that all three figures have equal bases and heights... Maybe you should just take a look at computing the difference between the images of the endpoints of the top of the squares to convince yourself.
The general form of such a transformation is
$\begin{bmatrix}1&a\\0&1\end{bmatrix}$ multiplying on the left of column vectors.)
Then you always have
$$[0,0]^T\mapsto [0,0]^T$$
$$[1,0]^T\mapsto [1,0]^T$$
$$[0,1]^T\mapsto [a,1]^T$$
$$[1,1]^T\mapsto [1+a,1]^T$$
From the first two you can see the length of the bottom horizontal line is $1$, and from the second two you can see the length of the top horizontal line is $1$. Obviously they also show the height was unchanged (since the $y$ coordinates were all preserved.)