Apple is warning users that the Back to My Mac service won’t be a part of macOS Mojave. This was already suspected as it hasn’t been a part of any of the beta versions of Mojave either. Now users are getting a warning dialog to make sure everyone knows.
Back to My Mac is a service that allows you to connect to another Mac you own and is set up using the same Apple ID. It got around the problem that people don’t necessarily know the IP address of their home computers, not do they have their home networking set up to forward sharing services. However, in practice, networking setups are so diverse and complicated with firewalls and settings that Back to My Mac never worked well for many people.
Back to My Mac only facilitated the initial connection, and then other services like File Sharing and Screen Sharing were used to accomplish tasks. These features aren’t going anywhere, just Back to My Mac. For instance, if you use Messages to request screen sharing, it will be unaffected. Also, many third-party screen sharing solutions exist, and Apple recommends using cloud services for remove file sharing. iCloud Drive is probably used far more today than Back to My Mac ever was.
If you purchase a new mac, will you still be able to connect it to your old mac?
glovideo: There are many ways to "connect" two Macs. Back to My Mac was an enhancement to some of those, though not the connection itself. By mentioning "new" and "old" Macs, I can assume you mean connecting using something like Migration Assistant. Back to My Mac has nothing to do with that.