What beneficial thing does eggshell do to tomatoes as a fertilizer? Does it heal leaf health, flowers, taking up nutrients, or something else?
Eggshells are a calcium source. Tomatoes need sufficient calcium in the soil to prevent blossom end rot.
Oyster shell and egg shells are a source for slow release calcium as they need to break down in order to make it available. Composition for both of these is mainly calcium carbonate crystals.
General usage of calcium soil amendments: If you determine you need calcium because the fruits are developing black rot on the blossom end, you will need to test the soil pH to determine whether you can add lime (Calcium Carbonate) which raises pH or add gypsum (Calcium Sulfate) which is more pH neutral. Lime helps make the soil less acid, gypsum is more appropriate for soil that is within the target pH range for the type of plant you're growing.