The Internet Check is used to determine if the access point has a working internet connection. If the Internet Check is enabled, and it detects that the internet connection has gone down, the access point will try to find a new route to the internet. This may mean a gateway turns itself into a repeater, or a repeater seeks out a new gateway. This may be preferable if you multiple access points connected to multiple internet connections.
When "Disable Internet Check" is enabled, it will turn off the Internet Check, which will keep the wireless network up, even if the internet has gone down. This is useful if you are planning on making a network that will not have Internet connectivity, spotty Internet connectivity, or you would like to keep access to local LAN resources even if Internet connectivity goes offline.