Homey uses the internet for a number of things, including:
Obviously, without internet, the above functionality cannot work. This means that Homey's functionality is relatively limited when there is no internet.
Offline Homey will still be able to manage your home, as Homey is not dependent on cloud services to operate (unlike some other systems). Functionality that remains available offline includes:
In short, all the capabilities that do not clearly require internet work. Homey will still be able to manage your home using your flows, and you will be able to control Homey via the smartphone app while at home. A lot of extra awesomeness is gone though.
Note: some devices (including most smart thermostats) require Homey to connect to them using the internet (and do not have a local workaround). Homey might not be able to control these devices while offline.