It's just someone else's computer made available to you!
OK, that's the simple version, but it's really not that much more complicated. It's similar to the difference between owning a car, and using a car service when you need to go somewhere. If you own your car, you have to pay for the car, pay for the gas and insurance, know how and when to keep it maintained, and replace the windshield wipers and tires and other parts as they wear out. But if you don't want to have to worry about all that, simply call a car service, tell the driver where you want to go, and then sit back and relax.
If you host your software in your own office, you have to buy the server computer, pay for the license, install hard drives, keep anti-virus and other software up-to-date, run your own backups, and replace parts when they go bad. If you don't buy a second server as a backup, then when (not if) your machine fails, everyone in your office that relies on the software on that server has to wait until a replacement machine is put in place.
With a server in the cloud, you are paying someone else to worry about the infrastructure upon which your software--and your business--rely. You simply access it over the internet!
We worry about the back end so you can focus on your business!