What if I don't do it?
All chains wear (this is sometimes called chain stretch). Dirty chains wear faster, particularly in gritty conditions. Once your chain reaches a certain level of wear, it will start to wear down the cogs on the motor and by the pedals (the sprocket and chainring). Eventually, they will wear enough that the chain will begin to 'skip' or the chain will simply snap from wear.
When the cogs wear down because of a very worn chain, you will need to replace both cogs and the chain. This is much more expensive than simply replacing the chain. Keeping the chain reasonably clean and replacing it before it becomes too worn will help to save a bigger maintenance job in the future.
You can check for chain wear using a standard bicycle chain checker tool.