Attracting a mate
It is possible that a female Carolina wren would choose a male based on the quality of the nest site the male has selected. Red-winged blackbird females, for example, choose mates based on the quality of the habitat the males defend. For wrens, producing multiple nests would give a greater likelihood that the male has found a site that a female will find suitable.
In reed warblers, males build two distinctly different types of nests. The females only raise young in Type I nests and Type II nests aren't constructed after the female lays her eggs, so the nests only play a role in advertising a male's fitness (source).
In the polygynous (one male will mate with multiple females) marsh wren, multiple males will build nests in a breeding area. Each male builds multiple nests and males will even continue to build "dummy" nests even after the females begin to incubate the eggs (source), making these breeding grounds a tangle of mostly abandoned nests. I could imagine a nest predator becoming increasingly frustrated after searching multiple nests and coming up empty (marsh and Carolina wren nests are deep cavities that take more than a passing glance to explore and see if they're occupied).
Avoiding competition/reducing parasitism
Egyptian geese aren't great nest builders, in fact they're aren't nest builders at all and typically lay their eggs right on the ground. Occasionally, however, they will harass and usurp nests sites from other birds, like black sparrowhawks (a large accipiter that nests throughout Africa). In areas where sparrowhawks had to contend with the geese, their reproductive success was cut in half (source). By building multiple nests at the same time, black sparrowhawks have an alternative path to raising their young if they're bullied out of a nest by a goose.
One more hypothesis (and the best one)
So here we are, as is often the case, without a clear reason behind why Carolina wrens in particular might build multiple nests. Marsh wrens, a close relative of the Carolina wren, build multiple nests to attract a mate. But they're polygynous, while Carolina wrens are monogamous. It seems less likely that a Carolina wren would need to attract the same mate year after unlike that site selection would have any bearing on mate selection.
One interesting peculiarity of Carolina wrens is that some female Carolina wrens lay their eggs in the nests of other Carolina wrens (source)!! It seems possible then that by staking out multiple nests sites, a bonded pair might avoid the energetic costs of having to rear another Carolina wren's young.