The books you're reading aren't talking about communications in the form of electrical signals, but instead about the main maritime routes between the two nations. In a military sense, your lines of communication are the routes which connect your outlying forces to their bases. In the case of Australia and America in WW2, the lines of communication were the convoy routes which ran from the east coast of Australia to Panama and California. The shortest route between these ran within aircraft range of the Solomon Islands. While convoys could be shifted onto longer routes, Japanese bases in the Solomon Islands could also be used to launch attacks on Fiji and Samoa. If these islands fell, then Japanese aircraft could hit any convoy between the US and Australia that didn't take extremely circuitous routes towards the South Pole. This would make shipping supplies and troops take so long as to be almost infeasible - by the time they'd gotten there, they'd be needed somewhere else.