Technically Matter itself doesn't require Internet access, but you'll come across many devices that won't operate (or even join) without a functioning border router. What's in the spec and the lived experience are sort of different here.
A border router does not provide internet connectivity.
A border router is used to bridge e.g. Thread networks onto a regular IP network so that your non-Thread smartphone can talk to it, and so Matter devices on different network types can link up directly if you want a Thread switch to control a WiFi light for example.
Matter devices use link-local IPv6 addresses and cannot route packets onto the internet even if they wanted to.
