What's the point of the first action in double-acting baking powder?

The first "action" is when it gets wet. This is the traditional baking soda + vinegar fizzing. The second rise in double-acting baking powder is when it gets hot.

In a baked item, such as a cake, this reaction to mosture helps create rise immediately, before the cake starts to change structure, solidify, and form a crust. This enables the center to begin to rise immediately. Without this initial rise, the outside of the cake could have already started to form a crust before the inside is heated enough to rise. This would result in "trapping" the center--either causing a denser unrisen center, or breaking/cracking the crust as it rises and escapes the crust.

The main rationale for an early reaction is to help create small bubbles during the final part of mixing that contribute to the fine structure of baked goods, as well as to add balance to the rate at which bubbles expand during baking.

Details:

There are practical chemical reasons for doing this: a lot of the standard ingredients used to make simple baking powders work through a sort of staged reaction. For example, Rumford Baking Powder uses monocalcium phosphate, designed to react precisely with the amount of sodium bicarbonate to avoid changing the pH in the batter. But its action happens in two phases, using that single ingredient:

Rumford Baking Powder contains only monocalcium phosphate as a leavening acid. Due to the nature of how this acid releases carbon dioxide gas with sodium bicarbonate in the presence of moisture, two-thirds of the available gas is released within approximately two minutes. It then becomes dormant at room temperature due to the generation of an intermediate form of dicalcium phosphate during the initial mixing. This stage of the reaction contains only one hydrogen ion and requires the catalyst of heat above 140 degrees F. in the batter.

Historically, this is how many baking powders worked. The first ingredients that tended to have a stronger delayed action tended to be aluminum based, and the movement away from aluminum products in the past couple decades has led to heavy marketing of "aluminum-free" baking powders. (Some people also find that such "slow acting" formulas leave more of a metallic/chemical taste.) The "aluminum-free" powders are often based on these historical formulas, which tend to release a higher percentage of gas at room temperature.

That said, there is still a benefit to having some first action of baking powders. The main reason is to create a consistent fine-grained baked product. As Shirley Corriher notes in Bakewise (pp. 46-47):

Baking soda and baking powder do not create new bubbles in a batter or dough; the carbon dioxide that they release only enlarges bubbles that already exist in the batter or dough. This means that bubbles created during mixing are a vital part of leavening, too.

Corriher goes on to note that liquids and eggs release steam that can also inflate the bubbles in batter, but once again, they do not create the bubbles. She concludes again by restating that "creating fine bubbles in the batter during the mixing is a crucial part of leavening."

Depending on the mixing technique, these bubbles can be created in a variety of ways, such as creaming butter and sugar together or whipping eggs. But the initial production of gas during the final part of mixing from baking powder reactions can be important in allowing these initial fine-grained bubbles to form as well. As noted here:

While the dough or batter is being mixed, it is important to generate sufficient incorporation of air. When this is insufficient, the dough or batter will not have enough air-cell nucleation sites for the development of a uniform grain structure. The resulting product can have a non-uniform grain, and this will risk blistering and tunneling problems in cakes and a tougher texture in cookies. Early leavening also contributes to early expansion during the baking process, helping to deliver sufficient volume. [...] Depending upon the product, the type of baking powder or leavening system can change to achieve certain goals. Many baking powders and leavening systems include monocalcium phosphate, monohydrate (MCP or MCPM) as a leavening acid. MCP reacts primarily during mixing and helps build batter viscosity and contributes to air-cell nucleation. MCP is known as a fast-acting leavening acid because about 60% of the reaction occurs within the first two to three minutes of mixing at room temperature.

Note in particular the bit about air-cell nucleation produced by baking powder. That link goes on to describe other types of baking powder options and their various effects in baking, depending on how much gas is released early vs. late.

It's sort of analogous to the rationale for kneading and reshaping bread dough after an initial rise: while you can make bread without doing this, part of the degassing and shaping is about breaking up large bubbles and creating a more fine-grained texture in the bread. But you need to have gas there to begin with as you are kneading/shaping to get maximum structure (hence the need for a first rise). Having baking powder release some gas during mixing serves a similar purpose, as it expands existing bubbles but further mixing can break them up to form smaller links in the batter and create a smaller foam. Ultimately, it will lead to a more consistent structure for the interior of the final baked product.

Also, baking powders used by home bakers are generally formulated to be multipurpose. As mentioned above, many cake batters, cookie doughs, etc. will include some other way of trapping gas during mixing, but not all recipes have a step like this. While many recipes will incorporate the baking powder at the very end with other dry ingredients (and just "mix until combined"), other recipes without other methods for trapping gases might mix the batter for a couple minutes at the end to use the gas generated from baking powder to leaven the dough and create the structure of bubbles. (One similarly sees recipes that use a combination of baking soda with acid along with baking powder: again, the baking soda reacts immediately and will help to trap gases and make bubbles even if the batter is just "mixed until combined," while the baking powder will enlarge those bubbles during baking.)

And there are baking powders (mostly used commercially) that produce the vast majority of their leavening in delayed action during heating. These are generally known as "slow-acting baking powders" rather than "double-acting." For example, Fleischmann's baking powder releases only about 10% of its gas initially at room temperature and 90% in the oven (compared to most standard double-acting powders, which tend to release 30-70% of their gas during the initial action). Slow-acting baking powders are more useful for batters and doughs that will be held before baking, hence their greater use in commercial situations. They wouldn't be as useful for recipes that depend on the initial gas release for a significant portion of bubble production during mixing.

The link quoted above also gives some specific examples of different types of chemicals used for baking powders (represented in the acronyms here):

To understand the application of these concepts, consider cake-style muffins. To create a product with a flat top and relatively dense and uniform grain, a single-acting baking powder would do the trick. MCP and soda in the leavening system or baking powder — a preblended combination of leavening acid and bicarbonate — would achieve the necessary incorporation of air during mixing. A flat-top muffin does not need as much leavening during baking.

On the other hand, for a bell-top muffin with larger volume, a double-acting baking powder with MCP and SAPP achieves the desired leavening effect. In the oven, the MCP would provide only 40% of its reaction, and the SAPP would contribute more than 50% of its reaction, giving the product more volume.

Lastly, to make a muffin with a peaked or cracked top, a baker could use a double-acting leavening system or baking powder with MCP, SALP and DCPD. Because SALP is slower than SAPP, the product would have gas produced for a longer time in the oven. DCPD would kick in very late in the baking process when the batter reaches a temperature of greater than 130°F (54°C), which would likely cause cracking in the product because the crust would be setting before the leavening was completed.

Each possible chemical in baking powder has a particular profile in terms of how fast it reacts at various temperatures. Emphasis on early and low temperature reaction will produce a product that is denser (but still uniform in structure), while exclusively using a product that won't react until high temperatures could risk cracking and might even reduce volume because the structure of the cake begins to set before the leavening reaction is completed.

Most general-purpose baking powders aim for a balance that both enhances early bubbles in the batter and produces gases to enlarge those bubbles and add volume during baking.