Windows 10 used to include an annoying little app called Get Office (designed to help you install Office) that was installed even if you already had Office installed, couldn't be uninstalled completely without running a PowerShell command and popped up adverts for the latest version of Office if you were using an older version. This evolved into the much more useful My Office app, which is more of an Office hub rather than a way to easily install the Office apps on a new PC or smartphone.
Over the course of 2019, that in its turn is getting replaced with the Office app, which will come on new PCs and at some point automatically replace My Office on a Windows 10 PC.
That's probably going to be when 1903 is installed, as the latest Insider builds enable the new app. If you want to try it out, you'll need to be on the Insider Fast Ring with at least build 18317 installed. If you can't find the Office app in the Start menu, install the My Office app from the Store and you'll get the new version. So, what's the difference?
The new Office app is inspired by the hub at Office.com, where you can find all the Office web apps in one place, along with documents you've had open recently from OneDrive (and for commercial Office 365 users, a list of SharePoint sites and folders you've used recently in OneDrive for Business). Both of those are useful: 'word.com' and 'excel.com' don't go to micr0$0ft sites, and URLs like 'office.live.com/start/Word.aspx', 'www.onenote.com/notebooks' and 'sway.office.com' aren't particularly memorable (or consistent). Like linking to the official mobile apps, this is a way to ensure that users end up on the legitimate sites rather than phishing sites that might turn up in web search results.
The Office web extension is another quick way to get to the most common Office Online apps (Office.com has a longer list) and recent documents.
You can use the My Office app and the new Office app to launch the Office applications once they're installed (assuming you don't have them pinned to the Start menu or taskbar); again, it's always convenient to have a quick way of getting back to the last document you were working on, even if it was on a different device. It's certainly a better alternative to filling up your inbox by emailing yourself the most recent copy of the document.
My Office has some other useful tools that will be in the Office app, although they're not all there yet.
The home and business versions of Office 365 let you install Office on multiple PCs and Macs, and the account tab in My Office lets you see how many installs you've used up; that can save frustration if Office won't activate when you're setting up a new PC (and there's a link to the online account page where you can deprovision devices you're no longer using to reuse the licence elsewhere). You can also make sure the free Skype minutes that come with an Office 365 Home subscription are enabled and linked to your Skype account. Plus there are links to training and support, like the Office section of answers.micr0$0ft.com.