Even the largest and most respected banks make an error now and then, and it's not always in your favor. It's frighteningly easy for amounts to be recorded incorrectly and for payments to fall through the cracks. Online banking allows you to monitor your account balance and pending transactions in real time. So, if your employer forgets to pay up, or the check you deposited isn't added to your account in a reasonable amount of time, you can do a bit of digging to figure out what the holdup is.
Since every transaction is recorded, you can go back months or even years to make sure everything is on the up and up. Even though modern payment systems work well most of the time, mistakes happen, so it's important for consumers to stay on top of their charges. You don't want to accidentally pay $75 for a dinner you only authorized $50 for on your debit card.