So to have a lunar eclipse you first need to have the Earth directly between the Sun and the Moon -- in other words you need to have a full moon as seen from Earth.
Here is a rough diagram of what a full moon might look like when seen from a disk Earth. I have drawn a disk of land and oceans and another disk of atmosphere above it, but one might have imagined the atmosphere instead as a dome or similar:
Notice that the Sun does not need to be aligned right above or beneath the plane of the Earth in order to be opposite from the Moon; in this case it hits the earth mostly from the side rather than from the bottom. To see this diagram the Moon needs to be both (a) full and (b) low in the sky; the full moon is necessary for any eclipse while the low-in-the-sky observations require a certain degree of luck in the timing of these particular eclipses that will involve non-circular shadows.
Now, this disk is still able to cast shadows on things, as is any piece of paper that you cut into a circle. Since the light source is coming from beneath the disk (it is night-time!) we would draw its shadow up and out of the disk-Earth like so:
Now the shadow here is not engulfing the Moon and that is no accident; not every full moon generates an eclipse but rather the moon can be displaced from the Sun, in this case in the third dimension (it is a little closer to you than the plane of the Earth and Sun and therefore the shadow misses it).
You will notice that there is a fuzzy edge to any shadow called the penumbra, and it comes from the light source having nonzero size -- if it were a point source (or infinitely far) then the shadow would have sharp edges; if it is a larger source (or closer) then the shadow has more blurry edges. In theory this could hide the entire effect, but in practice this blur is very visible during a lunar eclipse as the light which makes it through the atmosphere actually gets all of its blue color sucked out of it, so the boundary of the shadow on a lunar eclipse is seen to be a thin red line in every eclipse. So that basically tells you that this diagram is a little inaccurate in that the Sun should be much further away and the shadow being cast should be much sharper. In practice, you can make out the shape of the umbra when you are looking at a lunar eclipse, and you do not need to worry too much about the penumbra around it.
The only remaining factor is what that shape actually looks like. At these sorts of sharp angles where the eclipse is seen low in the sky, you would expect the eclipse to be a narrow ellipse. The lower in the sky the narrower the shadow should be, until you get an eclipse right at sunrise or sunset, when the moon is as low as can be across the horizon and the disk becomes a thin black line across the moon (plus a red penumbra on one or both sides).
So it might be hard to observe based on timing and coincidence, but sometimes you should be able to see the eclipse as narrower than the Moon is wide, so that you would have a band of darkness across the moon but with both sides of that band still illuminated. It would happen for eclipses just barely visible at sunrise or sunset in the opposite direction of the Sun.
But, this is not what is observed: lunar eclipses nearer sunrise/sunset when the moon is necessarily low in the sky, look the same as eclipses near midnight when the moon is straight above.
That is what Hawking is saying that the Greeks noticed and understood. The Earth must be "blown up" to a sphere or spheroid in this up-down dimension you are seeing, so that the Sun cannot hit it "edge-on" and thus cast a very narrow shadow onto the Moon during a lunar eclipse.