like earth and moon, they both orbiting on the center of mass of the whole system, that is earth and moon. However, since the earth is a whole lot more massive than the moon. The center is inside the earth. When you have a LEO satellite orbiting the earth. It is 2-body problem since the gravity of the moon and the sun don't impact the satellite much. However, when the spacecraft is on the lunar transfer orbit, things are very different, because the gravity of the moon become more and more dominant, (otherwise the moon cannot capture the sv to become a satellite of the moon). Then, it is not a 2-body problem any more. Earth, the moon and the SV here are the 3 bodies. Since the mass of the SV is very small compare to the earth and the moon, the effect on the orbits of the earth and the moon can be neglected. But the gravity forces act on the SV is drastically different from a LEO sat. Also, Sun's gravity also act upon the SV, and simple physics tells you that the resulting force F= F_e + F_m + F_s. When neither of these forces are significantly bigger than others, you can't ignore them.

That's where 3-body problem comes, (if consider F_s is comparatively small, we maybe able to ignore it, but that depends on the mission orbit).