Long since beaten, but the theory is that a katana is going to be useless indoors due to space, and so the wakizashi becomes your main means of defense when not outside. Since samurai were really an elite class, and spent a lot of time indoors (as opposed to outside farming all day), this is plausible.
However, the reality probably has more to do with the fact that it was a status symbol. Any footsoldier might have an old sword laying around, but to carry both swords through your obi was the sign of a samurai, and an indicator that you belong to a privileged class. I believe in later years (prior to being outlawed entirely) it was a punishable offense to carry both if you were not actually a samurai proper, but I’m not entirely sure.
Finally, there are valid nito (two-sword) fighting styles. In kendo, it is allowed (although rarely seen) to fight nito style, using a short shinai in addition to a normal one. The famous duelist Musashi was reportedly very good at this (using real swords, not kendo). In his case, the wakizashi would actually be used as a primary weapon along with the katana. However, it’s awkward and I personally doubt many people could have pulled it off. Basically, you have to get to a point where you can strike using one hand as powerfully and accurately as you could with two, which is a tall order…
And since people have brought up kendo being better/worse than fencing, I’d just like to point out that the two take completely different approaches to the philosophy of a duel, so it’s kind of a meaningless discussion. For example, in kendo you will probably never have electronic scoring the way fencing does, because there is much more of a focus on the proper dominance of your mind. A good technical hit may not score you a point if your zanshin isn’t convincing. As they say, “win, then attack.”
Of course, I don’t mean this as a slam against fencing in the least. I know some Japanese TV shows have done these face-offs in the past, and from what I’ve seen the fencer usually wins by a fraction of a second. If you rate simply by amount of time it takes to get off the mark and land a killing blow, fencing is probably going to win 9 times out of 10 because kendo strikes are cuts (with one exception). If kendo allowed more thrusts it might be more even, but then it wouldn’t be Kendo. 