How do packet cake mixes get their lightness without creaming butter and sugar?

Making a cake before around 1910 always required at least an hour of work beating sugar into butter: this incorporated air bubbles into the mixture. But since then, there have been a number of remarkable changes to the process.

The first was the arrival of modern cake shortening, which has smaller fat crystals that trap small air bubbles, which stay in the batter. They also have a much wider temperature range at which it has the right solidity, and these days manufacturers fill shortening with preformed bubbles of nitrogen (around 10% of the volume) and up to 3% of the shortening weight is replaced by bubble-stabilizing emulsifiers.

The second major innovation was the arrival of cake flour: a soft flour low in gluten that is very finely milled; in the US it is strongly bleached with chlorine, which also lightens the end result by various means, but the EU and UK disallow the bleaching step, so manufacturers there use different processes such as heat treatment.

These two improvements, together of course with mechanical mixers, have lead to not only less work when making cakes but also lighter, more fluffy end results, and the ability to use more sugar without the batter becoming to heavy.

Finally, because the mixing process fills the batter with bubbles, cake recipes typically either for no chemical leavening or for less than other batter recipes do.

_{(All of this information is extracted from pages 555-557 of the excellent On Food and Cooking by Harold McGee.)}

So the answer to your question seems to be: mass-produced cake mixes use the incredible advances in food technology of the twentieth century in general, not so much chemical leavening specifically.

Cakes pretty much always have chemical leaveners in them, whether you're making them from a mix or from scratch. So yes, there are chemical leaveners, and that's plenty to get it that light. I'm not even sure if it takes significantly more leavening than you'd need if you'd creamed butter and sugar, since cake batters are wet, so sugar will generally dissolve more and not provide a structural component. Some cake recipes don't even bother with a creaming step. In any case, it obviously works. Creaming butter and sugar is perhaps more of a factor in cookies, as mentioned in this question, for example.