You have a SharePoint Online document library that requires checkout and you have it synced locally via OneDrive for business. Files opened from either online or from the local synced folder open documents read only with no way to check in or check out files from the Office client.
This is a current design limitation with OneDrive,
"Libraries with Checkout, Required columns or metadata, or when Draft Item Security is set to either Only users who can edit or Only users who can approve items in Version Settings of the library."
- Do not sync SharePoint document libraries that require checkout
- Uncheck the OneDrive setting: Office[tab] \ "Use Office 2016 to sync Office files that I open"; If you uncheck this global setting, co-authoring will not work for any file opened from a local synced folder. But co-authoring will work for any files opened from online. GPO for controlling this setting is EnableAllOcsiClients defined at: https://support.office.com/en-us/article/Use-Group-Policy-to-control-OneDrive-sync-client-settings-0ecb2cf5-8882-42b3-a6e9-be6bda30899c#enableallocsiclients
Supporting libraries that require checkout is on the product groups backlog and maybe fully supported at some point in the future.