Because they match up the physical units of each element, satisfying the conservation of mass.
The most important chemical discovery, in my opinion, was the concept of the mole. The understanding that ALL elements consist of the same number of ‘reactive units’ of differing masses. This understanding is what allowed us to to take observed changes in reactant and product masses and relate them to the elements involved in the reactions. Thus, “balancing” a reaction is the key to knowing exactly HOW elements will combine in a reaction, and what the form(s) of the products will be.
They work by ensuring that the number of moles of each element in an equation are equal on both sides of the reaction. Physically the compounds may not be perfectly balanced (and usually, they are not), but the balanced chemical equation can then let us calculate just how much of each reactant CAN be reacted to a product. So, we can start with the name, and then the formulas for any particular set of compounds. Writing them out into our “equation” form specifies what we expect (or know) the products will be.
THEN we do the “balancing” to ensure that the ratio of moles (elements) is correct. Finally, we can use the balanced equation to determine how much reactants and products we need for a complete reaction. We can also then relate things like the energy (given off or required) used in a reaction to a particular combination of compounds.