How do we know that living things are made of cells?

In Latin they had a word, scire, "to know". The noun form of that word is "science" and all science is defined by the scientific method:
1. Observe something.
2. Formulate a hypothesis.
3. Devise a test.
4. If the test fails, go to #2.
5. If the test passes and is confirmed, the hypothesis might be promoted to a theory and used to prove other hypotheses. And it might not.

We know living things are composed of cells because we looked, and we saw something we decided to call cells. The name of the man who first looked was Robert Hooke.

Robert Hooke, an English scientist, discovered a honeycomb-like structure in a cork slice using a primitive compound microscope. He only saw cell walls as this was dead tissue. He coined the term "cell" for these individual compartments he saw.
Cell Theory Timeline - Soft Schools
www.softschools.com/timelines/cell_theory_timeline/96/

Anton van Leeuwenhoek is another scientist who saw these cells soon after Hooke did. He made use of a microscope containing improved lenses that could magnify objects almost 300-fold, or 270x. ... Leeuwenhoek named these “animalcules,” which included protozoa and other unicellular organisms, like bacteria.
Cell theory - Wikipedia
en.wikipedia.org/wiki/Cell_theory

Before Leeuwenhoek made his microscope, nobody knew where babies came from except that you had to have a woman. He examined his own sperm and found cells in it. That was about 1670.
en.wikipedia.org/wiki/Antonie_van_Leeuwenhoek