Light through the glass
Simulator
Both panels are the same lens at the same moment. Like any ray diagram this one is drawn to two scales, stretched across the axis and squashed along it, and the aperture is drawn smaller again so that a long fast lens still fits the panel. What is true to the numbers is the difference between the two views.
One lens, two focal lengths
A spherical lens is the same lens whichever way up you hold it: every surface is part of a sphere, so light is bent by the same amount in every direction. Put one cylindrical surface in the path and that stops being true. The lens now has one focal length across the frame and a different one up it, and every strange and desirable thing about anamorphic follows from that.
A cylinder has power in one axis only
Lay a glass rod down and look along it: across the rod it is curved and bends light, along it the surface is straight and does nothing at all. That is the whole trick. In the view from above the cylindrical elements are lenses; in the view from the side they are flat slabs the light walks straight through.
The horizontal field of a shorter lens
A 2× anamorphic marked 40 mm has a vertical focal length of 40 mm and a horizontal one of 20. Up the frame it sees what a 40 sees, across the frame what a 20 sees. It is not wide and long at once by magic, it is two focal lengths in one barrel.
The sensor shape decides the squeeze
Desqueezing multiplies the sensor's aspect ratio, so the number you want depends entirely on what shape the sensor is. A 4:3 sensor wants about 1.8× to land on 2.39, and 2× overshoots to 2.67 and gets cropped. A 16:9 sensor wants about 1.33×. That arithmetic, not fashion, is why the mild squeezes exist.
Why the ovals are what they are
The iris is round and the cone behind the lens is the same in both axes, so the blur would be round, were it not that the horizontal magnification is smaller by the squeeze. Defocus scales with the square of magnification, so the blur arrives squeezed by the square: four to one on the sensor at 2×, and still two to one after the picture is stretched back out.
The pupil is wider than it is tall
An attachment that compresses the field widens the beam by the same factor, so the hole the lens looks out of is an oval lying on its side. A bright source reflected off a cylindrical surface spreads along the axis with no power, which is the horizontal streak, and the wide pupil is why it reaches the far edges of the frame.
Front and rear are not the same lens
An anamorphot in front of the taking lens squeezes the world before it is imaged, and focusing it moves glass that also shifts the squeeze a little, which is where close-focus mumps come from. Squeezing behind the iris holds the ratio at every distance, and asks the spherical group in front to cover an image circle wide enough to be squeezed down afterwards.