https://wiki.tfes.org/The_Lunar_Eclipse describes the FE position rather well.
I'm not an expert on all of the machinations of FE (primarily because I find it hard to suspend disbelief), but what they appear to be saying is that there is another object in some orbit around the Earth (I have no idea how it could orbit a 32 mile wide Sun; the 2-body system would be quite unstable). This object causes eclipses by moving in front of the Moon and the Sun.
I'm not going to start a debate here, but if you want to know the Round Earth camp's position on this, start a thread in the Flat Earth Debate section and a lot of voracious sharks will jump on this one, since it's so easy to challenge.