Modern scientist can do simulations and stuff, but long ago people could figure that out relatively easily.

People write down stuff like seeing a solar eclipse it is pretty memorable.

If they write down when they saw an eclipse it gets relatively easy to see a repeating pattern.

If you pay a lot of attention you notice things like all solar eclipses happening during a new moon and all lunar eclipses happening during a full moon.

Next you notice that sometimes the moon during a new moon is above the sun and sometimes below it and that it only gives you an eclipse when it is directly in front of it. You can watch the moon move up and down in relation to the sun and you figure out that there are two repeating cycles involved, one of the phases of the moon and one of the moon going up and down. When both the moon going through a node and the moon going through a full or new moon coincide you get an eclipse.

It doesn't take much to predict when there will be a new moon and a full moon many years in advance and figuring out the other bit also just requires a bit of paying attention.

Then the rest is just a bit of math.

There is an 18 year cycle (223 times the moon through its phases) after which the pattern for possible eclipses repeats. So once you know about enough eclipses happening in the past you can just extend the known pattern forward even without understanding the physics or geometry or math behind it.

The Eclipse today matches the one that happened on October 14 in 2004 and the one that will happen November 4, 2040.

You can just always go backwards or forwards 18 years and 11 days (plus or minus a day depending on how many leap years there were).