First, this might be a better question for /r/asksocialscience

Disclaimer: I don't know of peer-reviewed studies on the issue, so if anyone out there knows of any please let me know.

As /u/terrorbites mentioned, there are still places in the world where fat=beautiful or powerful. He mentions Bangladesh, I've also seen it in Kenya and Tanzania.

This source has some very informative maps, depicting what percentage of their income people spend on food in various parts of the world. The U.S. is the lowest at 6.4%, while some African and Asian countries are over 40%.

Now, the following stands to reason:

1. Being fat means you can afford a lot of food, and
2. Being able to afford a lot of food is going to be a much bigger symbol of wealth and power when most people spend 40% or more of their income on it

Like I said at the beginning, I've never seen a peer-reviewed study on this.

That said, you can basically correlate the end of fat=beautiful or powerful with a country's economic development. After food gets cheaper relative to average income, being able to afford a lot of food is less impressive...when food is expensive, only the elites are fat. When it's cheap, anyone can be fat.

Once you make the turn into the 19th century European/American fashion is no longer cut to accentuate the larger figure (for men). Even in the 18th century you see the waistcoats being cut to accommodate, and sometimes even accentuate, the belly. Going back further to the 16th century there is even padding in doublets to create this illusion.

The 19th century features a slimmer cut with a narrow waist and more accentuated shoulders. It's a bit unrealistic, but this fashion plate from 1848 shows how the "bulge" moves upwards to become a barrel chest, rather than a fat stomach. The accent remains in this area for a while longer, flattening out at the turn of the century.

For women a curvier figure continues in popularity until the 1910s. 1890s corsets were still cut for curves, while the 1910s corsets began to slim the hips. Thin is popular throughout the 1920s, goes out again by the 1950s, comes back in the 1960s, etc.

Being fat is just another body type during these times, though. It just isn't always the fashionable body type. It isn't until the later half of the 20th century, when food is readily available to almost all, that an excess of weight is seen as a negative body shape.