His work on famines first came to wide attention with his 1981 book Poverty and Famines.

As a New York Times article from 2003 notes:

When the International Labor Organization asked him to look into the causes of famines in the mid-1970's, Mr. Sen decided to focus on the Great Bengal Famine of 1943, in which as many as three million people died. As a 9-year-old boy in a privileged Bengal family, he had seen the suffering first hand. At the time of his research, it was widely assumed that famines were caused by sudden food shortages. Examining records, however, Mr. Sen found that food production in Bengal had not declined. Rather, food prices had soared while farm wages had sagged, making it hard for rural workers to buy food.

Examining more recent famines in Ethiopia and Bangladesh, Mr. Sen found that they, too, were caused not by food shortages but by lagging rural incomes. In his landmark ''Poverty and Famines'' (1981), he argued that most famines could be readily prevented by mounting public works projects for those most in peril.

That book did not consider the role of democracy. Soon after it appeared, however, Mr. Sen began hearing reports about the Chinese famine of 1958 to 1961. The full dimensions of that calamity had remained hidden from the outside world, but after Mao's death it became clear that tens of millions had died. To Mr. Sen the reason seemed clear: the absence of a free press and opposition parties meant there was no one to sound the alarm. By contrast, India had been free of famine since independence. In a 1982 article for The New York Review of Books, Mr. Sen argued that even a fraction of the Chinese death toll ''would have immediately caused a storm in the newspapers and a turmoil in the Indian parliament, and the ruling government would almost certainly have had to resign.''

In Bengali famine of 1943, for instance, Sen argues pretty convincingly that food produced in Bengal was much the same 1943 (a famine year) and 1941 (a non famine year). Previous arguments had always assumed that famines were supply side problems, that is, there wasn't enough food, therefore the assumption was that the difference in the two years was that there was more "carry over" in 1940 than 1942. Sen, however, argues that it is demand side that matters. Particularly, he argues it is not a lack of food [1941 was like 1943, and they're even similar when considering two or three year averages to look for expected differences in carry over], but a problem in the distribution of purchasing power—namely, that the War economy both increased demand and especially left behind rural wage earners as urban wages rose, making rural wage earners the primary victims of the famine. It's not that there wasn't enough food, it's that the price of food shot up, putting out of reach for three million [!] in pre-partition Bengal.

I don't have the book on me (it's one of those I picked up and never finished), but you can get a pretty clear idea of his argument by reading a series of exchanges in the New York Review of Books. The first is Joseph Lelyveld's review of Madhusree Mukerjee's Churchill’s Secret War: The British Empire and the Ravaging of India During World War II ("At first glance, Mukerjee’s book is a detailed working out of Amartya Sen’s proposition that a democratic government will preserve its people better than a regime that rules from afar") [most of the following quotes from these letters, which I have handy, not Sen's book, which was long ago returned to the library]. This devolved into 3 rounds of letters (1, ["Joseph Lelyveld notes that I do not discuss Amartya Sen’s assertion that Bengal contained enough grain to ward off famine. I avoided this aspect of Sen’s work because his conclusion of sufficiency in Bengal has been seriously challenged. Historian Mark Tauger has shown that Sen based his crop estimates on projections, and that crop diseases spread by wet weather appear to have drastically reduced the actual harvest."; Sen replies in part, "Mark Tauger’s data come from exactly two 'rice research stations' from two districts in undivided Bengal, which had twenty-seven districts. Since weather variations have regionally diverse effects, it would require more than this to 'seriously challenge' the analysis I made, using data from all districts, which indicated that food availability in 1943 (the famine year) was significantly higher than in 1941 (when there was no famine). Ignoring the range of data I used in my study, she misdescribes my estimates as being based only 'on projections'" and "There was indeed a substantial shortfall compared with demand, hugely enhanced in a war economy, as I have described in detail, but that is quite different from a shortfall of supply compared with supply in previous years."] 2 [Tauger says biologists at the time thought there was a famine based on supply, Sen says, no, but seriously, I have better data than the biologists], 3 [Tauger replies in part, "Sen either failed to read what I wrote or chose to misrepresent it," Sen then gives a pretty good case for why his data and arguments are better). For those interested in what academic arguments look like, read the letters. Others should, skip to Sen's part of the last round of letters. But basically, he argues that the short fall in supply in 1943 was common, having occurred in previously in 1928, 1936 and 1941. In 1941, for example, a slightly more severe shortfall caused a no famine. Why was 1943 different? Not a shortage of supply, but changes in demand, particularly the boom in urban wages which lead to high prices in cities and famine in the countryside.

His letter ends:

different economic influences [fueled] the boom economy of Bengal [in 1943], with a sharp rise in the demand for commodities, particularly demand for food, but which also left some people, especially rural wage earners (whose wages had lagged seriously behind food prices), without the means to obtain much food.

The upward pressure on food prices was enhanced by the government’s drive to procure rice from rural Bengal to feed the Calcutta population through making food available in urban ration shops at heavily subsidized prices (mainly for helping the successful conduct of the war, for which “peace in Calcutta” was taken by the Raj to be extremely important), as well as by the speculative withdrawal of food stocks from the market by professional traders and also by the panic-stricken public. The sharp increase in food prices had devastating effects on the rural poor, especially on rural wage earners.

I focus on the 1943 argument because that's the chapter I read most closely in Sen's book [it was the letters in the NYRB that sent me to the library], and only really glanced at the others, but it's also where Sen's thinking about famines begins and always seems to return to, especially this comparison between the same supply in 1941 and 1943 with different demands. Sen writes in his original volume, “No matter how famine is caused, methods of breaking it call for a large supply of food in the public distribution system.” Sen is not arguing that famine has no natural influence [in other parts of his book, he discusses famines caused by a much sharper drop in supply, like Ethiopia], but rather famine is always preventable, and therefore, any failure to prevent famine is a failure by first the market (distortion in wage and price) and then by the government (who failed to remedy this by public distribution). That's why, as Sen argues, there's never been a famine in democracy. There's been, and still is, hunger. There are even deaths by starvation (as that first Times article points out, there are sometimes a hundred starvation deaths a month in some Indian provinces), and Sen himself has elsewhere argued that four million people die prematurely in India every year from malnutrition and related conditions, but there has never been a famine because a free press and democratically accountable politicians simply wouldn't stand for it. Sen's argument has been criticized from the left (who argue that his limited view of famines and limited public distribution to prevent them ignore both larger problems, like malnutrition and land reform) and right (who argue that free markets prevent famines and things like economic liberalization is the key to increasing both food supply and purchasing power). Others argue with parts of the data he relies on, saying we should ignore it as unreliable (a whole section on the Bengal famine wiki makes this point; while it doesn't make this connection until later in the article, this section seems aimed at Sen; and the substance of Mark Tauger's letters to the New York Review of Books were based on a different set of criticism).

While I'm not confident enough to say that no famine was caused by natural causes (I'm not a famine expert), Sen does make a very convincing case that many devastating modern famines are not caused by an total lack of food, but rather problems in the distribution of said food, often caused not primarily by changes in supply, but changes in demand, which lead to things like hoarding and some people starving while others do fine. A cyclone or a blight is not enough to cause famine alone. During the potato blight, for example, Ireland was famously still a net exporter of food to England. For a variety of products, exports actually increased during the Great Famine.