All the planets in the solar system orbit the Sun roughly in the same plane. That means that from the side, the orbits of the planets would all be very close to falling on the same line. Since we orbit the Sun on the Earth, we're on that line, too.
As it happens, Jupiter's poles are almost perfectly perpendicular to the line. In other words, its equator lines up with the plane of the solar system... and its big moons also orbit the planet right above the equator.
All this together means that when we look at Jupiter, the moons appear to orbit the giant planet on a line, too. They swing back and forth, moving "left to right" and "right to left" over time. In fact, if you watch Jupiter for long enough you'll certainly see a moon pass directly over the planet's face, and sometimes you can see the moon's shadow as well. It's very cool.
That. Is. So. Cool.
You can see the shadow of Io (which is roughly the size of our own Moon) pass over Ganymede (as big as Mercury!), then Io itself pass directly between us and the giant moon. Not only that, but take a closer look at Ganymede: you can see surface features! That's truly astonishing. I can almost swear I see similar features on Io, but the distortions due to Earth's atmosphere on the image make that a very difficult thing to be certain about.
I find it amazing that we can see things like this. Imagine! Christopher had software that was so accurate in its mathematical and physical modeling of the solar system that it could predict this transit. He had a telescope and a digital camera powerful enough to record it. And, of course, we have the intertoobz which allows you to see it.
In my line of work (y'know, truth promotion) I hear from people who think science is all guesswork. "Yeah, but how do you know?" they ask. The answer is: math. And physics. And chemistry and optics and engineering and Kepler and Newton and Einstein. We know because we test our assumptions, and if they don't hold up they're gone. We keep the good stuff, the stuff that's proven itself. And eventually we get models that are so good they can predict when and where two objects hundreds of millions of kilometers away pass in front of each other.
Yeah. That's how we know.
Man. I love this stuff.