Think of your chain as a consumable like oil in a car, if you do not replace it when you should, it starts to prematurely wear on other parts. In the case of your chain, it starts to prematurely wear components like your chainrings and cassette.
Your cassette and chainrings are expensive so by replacing your chain when you should, prolongs the need to purchase a new cassette saving you money in the long-run. Worn chains start to slip and sometimes skip cogs, reducing your performance and your cycling experience. The chain brand, riding conditions and your riding style all play a part in the wear and tear of your components.
Here is a great video to show how to check when you should replace your chain. The Hubtiger app tracks the distance you have done on your components, including the chain so you can proactively service your bike on time, saving you money in the long-run.