Many ocean organisms live only at specific depths. For example, the coral Acropora palmata only grows within a few meters of the sea surface, where it can get the most sunlight.

Suppose you drill a core into a coral reef using a floating drill, and find some of this coral that's 10,000 years old at a depth of 100 meters below the surface. How could this happen? Either the sea level must have risen, or the island the coral is growing on must have sunk.

You can figure out whether the island has sunk by looking for very old corals of the same type. For example, if you find corals dating back to a previous warm period 120,000 years ago at or above sea level, you can be sure that the land isn't sinking very fast.

You can repeat this analysis for other corals that live at greater depth, you can repeat the analysis in coral reefs all around the world, and repeat it for other types of organisms that live near the water surface in colder climates, and you find a consistent pattern of sea level change over time.

> Is someone who denies the historic sea level graph equivalent to a Flat Earther?

There aren't many people who deny that sea level has changed. Most often climate change deniers use the past sea level graph to show that sea levels are not stable, and thus -- they say -- the current sea level rise is nothing special, and a natural process. Some people argue that the ages of the changes are wrong, and the sea level changes actually represent the biblical flood.

http://people.uncw.edu/grindlayn/GLY550/Fairbanks-Sealevel-1989.pdf

https://www.sciencedirect.com/science/article/abs/pii/S0012821X98001988

https://commons.wikimedia.org/wiki/File:Post-Glacial_Sea_Level.png