The evidence is based on the assumption that Lunar Eclipses are based on the shadow of the Earth falling on the Moon. This assumption stems from the fact that Lunar eclipses only occur on a Full Moon, and that they follow a pattern related to the angles between the Sun, Moon and Earth, angles which can be shown vary in a predictable way.
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We can see that the shadow of the Earth is ALWAYS curved (and actually almost circular), regardless of which part of the Earth the moon is visible from. The only shape which give the same shape shadow from every direction is a sphere (or close to it).
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Because of the irregularities on the Moon’s surface, it would be very difficult to accurately measure the Earth’s oblateness (the degree to which it is non-spherical) just from observations of the Lunar Eclipse.
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The best way to measure the oblateness is through accurate survey of the surface; especially since the difference between the equatorial circumference and the Polar circumference is so small.