Users of those older applications will not be actively disconnected, but no effort will be made to accommodate access after Oct.13.
No support also means that Office 2010 and Office 2016 for Mac users connecting to Office 365 online services like OneDrive and SharePoint may, over time, see those services become unreliable and eventually unavailable. While users may continue to use applications found in those productivity suites, over time bugs and vulnerabilities are likely to result in reliability, compliance, and security issues-issues you will have to deal with on your own. This seems like a firm date and is not likely to be extended by Microsoft because of customer complaints.
13, 2020, Microsoft will no longer support Office 2010 and Office 2016 for Mac.