Prior to Office 2016, "Send to OneNote" ran as a separate exe with its own window. In 2016 it is a windowless exe that runs in the system tray (the place on the desktop with all the icons, next to the clock). This should be running by default, you can check your system tray:
If you don't see it, open OneNote and navigate to File >> Options and select "Display" and make sure "Place OneNote icon in the notification area of the taskbar" is checked:
To configure where stuff goes when sent to OneNote, go to the "Send to OneNote" tab:
If you right click on the tray icon, it will bring up the keyboard shortcuts available:
The thing that's missing from previous versions of the app is the button that would "Send to OneNote" from various apps (IE, Word, etc). Apparently this is now the responsibility of the other applications instead of being controlled through the "Send to OneNote" tool (i.e. you need to click the appropriate "Send to OneNote" button in the other apps).
Here is a "User Voice" suggestion to bring back the previous incarnation of "Send to OneNote"