There are a number of reasons why chemistry students study stoichiometry. I'd say the most important is the ability to make useful predictions.
Stoichiometry allows us to make predictions about the outcomes of chemical reactions. Making useful predictions is one of the main goals of science, the other being the ability to explain phenomena we observe in the natural world.
So what kind of predictions can we make using stoich? Here are some examples:
Predict the mass of a product of a chemical reaction if given the starting masses of reactants.
Predict the volume of a gas which will be produced by a reaction if given the starting amounts of reactants.
Determine the optimal ratio of reactants for a chemical reaction so that all reactants are fully used.
Stoichiometry is also challenging, so there are lessons about work ethic, problem solving techniques and perseverance...
Stoichiometry gives us the ability to tie together learning about chemical reactions, formulas of compounds, mole calculations and conversions!
Here is a video which uses the analogy of making sandwiches to introduce the concept of stoichiometric calculations.