As a quick disclaimer: I approach this primarily from the American home economics movement, which was heavily about nutrition, not as a food historian, so if we are lucky someone else can give you a more culinary perspective, but here is the sociological perspective.

“Digestibility” at is right at the birth of nutrition as a science, and “digestibility” is a polite advertising buzzword to intimate to you, dear homemaker, that the food will not give you a tummy aches, heartburn, or toots. And heck, we still get food advertising to that effect. Humans in general before this time had a fair idea that food goes in, it churns around in your stomach, poop comes out, both of these happening regularly and in the least painful ways possible is preferred, but a lot of the details about how exactly that works get worked out (perhaps not to our understanding today, but still worked out to themselves) around the turn of the century. For example, most vitamins were discovered from 1910-1930s, and how exciting would those breakthroughs have been? Consider this cutting-edge nutrition book from 1923, which will help you get more of all FOUR vitamins in your diet. Nutritional science is hot though, and of great interest to many parties: poor parents, who have agonized watching their children suffer the effects of poor nutrition, especially during the Great Depression; rich parents, who also stress over their children and if they’re healthy; governments, who have a pretty hefty interest in maintaining a healthy population for various purposes, especially after WWI; and Progressives, rich charity-minded folk who care about the poor and want to better their lives. Enter the food producer, who wants to market to all these people. This concern for nutritional science starts entering the consciousness of homemakers as they get exposure to home economics training through school, charity efforts, and government programs, from the turn of the century through the 50s. Any homemaker worth her (iodized, of course) salt is going to care that her family is fed nutritious food, that gets in and out of their bodies with the least drama. Advertisers cater to her scientific mind.

So what makes a food “easily digested?” Low fiber, first, cause fiber gives you the toots and that was not seen as a part of healthy digestion then and nobody really likes it now even, but also low fat, low acid, well cooked (proteins and stuff broken down), and now spicy foods are banned in the bland diet too, but that wasn’t much of a concern for a White American housewife at the turn of the century. So some advertising, like this one for canned beans, makes sense. Cooking the tar out of your beans will indeed make them more “digestible,” and maybe you won’t fart as much, especially if Jane Homemaker doesn’t know all the tricks to cooking beans to make them less noisy. Early breakfast cereal, also makes a fair pitch to being digestible, because all the wheat is pre-cooked and broken down. However, that “digestible” word is slapped on a lot of things that, uh, aren’t really more easily digestible than anything else. Like your Worcestershire sauce there. I spent a while trying to find any “science” the company had put behind that claim, but I’ve come up a blank. You didn’t need to prove a word like “digestible” back then though. I think it was a general grasp for the “health halo” effect, like when they market potato chips as “natural” now.

Probably the most famous marketer under the “digestible!” banner is Crisco, who found a pretty good way to market a rejected soap-making fat in their own bestseller, The Story of Crisco, where they give us this very [citation needed] claim

The first step in the digestion of fat is its melting. Crisco melts at a lower degree of heat than body temperature. Because of its low melting point, thus allowing the digestive juices to mix with it, and because of its vegetable origin and its purity, Crisco is the easiest of all cooking fats to digest.

When a fat smokes in frying, it "breaks down," that is, its chemical composition is changed; part of its altered composition becomes a non-digestible and irritating substance. The best fat for digestion is one which does not decompose or break down at frying temperature. Crisco does not break down until a degree of heat is reached above the frying point. In other words, Crisco does not break down at all in normal frying, because it is not necessary to have it "smoking hot" for frying. No part of it, therefore, has been transformed in cooking into an irritant. That is one reason why the stomach welcomes Crisco and carries forward its digestion with ease.

Also choice quote:

Crisco is Kosher. Rabbi Margolies of New York, said that the Hebrew Race had been waiting 4,000 years for Crisco. It conforms to the strict Dietary Laws of the Jews. It is what is known in the Hebrew language as a "parava," or neutral fat. Crisco can be used with both "milchig" and "fleichig" (milk and flesh) foods. Special Kosher packages, bearing the seals of Rabbi Margolies of New York, and Rabbi Lifsitz of Cincinnati, are sold the Jewish trade. But all Crisco is Kosher and all of the same purity.

4000 years.

Later Crisco got more explicit about what they mean with “digestibility:” it’s all about preventing getting heartburn from all the fried trash you eat.

The aperitif/digestif thing in alcohol is a different beast though, and outside of nutritional science/home economics. Most home ec scientists were Prohibitionists and teetotalers!

Tl;dr: it’s usually code for “won’t give you heartburn,” or farts, or general indigestion, a concern which isn’t advertised to as much in food anymore, because another industry got in on that gig (see: Beano, Zantac, Pepto Bismol, etc etc)

One thing I'd like to point out here that hasn't been mentioned (and isn't in /u/caffarelli's otherwise excellent answer) is that prior to the modern industrialized age being infected with Helicobacter Pylori was quite common, and nearly universal. It still is in the developing world. The changeover likely having a lot to do with improvements in sanitation (especially in drinking water, but also foods, especially preserved foods). In any event, H. Pylori infections are known to cause inflammation of the stomach lining, gastritis, ulcers, and stomach cancer. Probably about 10-20% of people with H. Pylori infections develop peptic ulcers. In the developing or pre-industrial world something like 80% of people will be infected by H. Pylori by their mid-20s. Which means that somewhere around 1 in 12 to 1 in 6 of all people in the pre-industrial world would end up with a peptic ulcer at one point in their lives.

That's a lot of people. On top of that you'd have random bouts of acute gastritis or less severe forms of just having a more "sensitive stomach" than usual. All of those symptoms would have lead to a desire to alleviate or ameliorate them. Experiencing a peptic ulcer even once is fairly likely going to lead to a lifetime of being careful with what one eats in an effort to avoid it, which will encourage others to follow the example out of an abundance of precaution. And even if the behavior has no effect often times the gastritis or ulcer will simply get better naturally, leading to the perception of a causal relationship.

You might also want to ask this over at r/AskFoodHistorians/.