Every game is a fight over control of the board. Control is often gained or lost by players exploiting each others connections. Making sound (as in healthy) connections means making connections that are less likely to be exploited.
As a counterexample, consider the unsound connection below.
Blue can only rescue two of the three bridges.
Soundness is not only non-overlapping connections. It is also the practice of grabbing territory when given the chance. For example, when joining two groups of pieces there are often several different moves which join the groups but claim territory in different ways.