They host the data, not the application. Read carefully. Office 365 description from MS site:
Microsoft wrote:
You are and your business are on the move. That means you need to be able to work from anywhere, across your devices, and have a consistent and fast experience. That freedom is what you get with Office 365. Whether youre at your desktop or on your tablet or phone, you can get to your documents, email and calendars, shared files, and online meetings. And wherever you access it, its always your Office, so your files are up to date and your recent documents just a click away.
The 'app' since this is what we call software now, resides locally but the data is remote.
Microsoft wrote:
Let Office 365 take care of IT, so you can spend your time moving your business ahead. Install Office quickly and set up users with step-by-step guidance.