Origins are in contradiction between 'excess' food production and hunger/want. Early days of Roosevelt New Deal included purchasing crop surpluses and distributing them to local agencies that made 'relief packages' available to 'needy' families. Department of Agriculture (USDA) established first food stamp program in 1939 (some accounts attribute idea to then-agricultural secretary Henry Wallace who would become Rosevelt's vice-president in 1940, be dumped by Democratic Party in 1944 in favor of Harry Truman because he was thought to be 'soft on communism' and run ill-fated campaign for president on Progressive Party ticket in 1948). About 4 million people participated in program that was eliminated in 1943 amidst WW2 reduction in both unemployment and surplus food.
Current (more or less) food stamp program was enacted in 1964. USDA, motivated only incidentally by desire to feed the poor, had begun another distribution program in 1950s when 'problem' of excess food production returned. Some liberals in Congress began talking about hunger and introducing food stamp bills in late 1950s. John Kennedy promised both increased farm supports and food stamp program in his 1960 presidential campaign. As in 1930s, it is unlikely that food assistance would have been considered absent large crop surpluses.
Initial number of food stamp recipients was about 500,000 and appropriation was about $30 million. Program growth was minimal for several years but 1968-69 witnessed increase to over 3 miilion recipients and about $230 million in spending. Why latter growth? Poor became increasingly militant. Much too short-lived National Welfare Rights Organization (NWRO) and numerous similar local groups began engaging in protest and disruption (mainstream social science literature calls this stuff 'client-center mobilization').
Agitation was intended to improve welfare benefits *and* to remove stigma attached to welfare that led folks who were eligible to choose not to apply. Food stamps were part of Poor People's March agenda that Martin Luther King, Jr was organizing at time of his assassination (King was killed because of his leftward trajectory - public opposition to Vietnam War and growing economic egalitarianism, he was in Memphis supporting striking sanitation workers on night he was gunned down). While the march itself did not meet expectations, 'Resurrection City' was built opposite Ag Dept building in DC and USDS was principal target of protests.
Repeated use of direct action produced policy benefits. By mid- 1970s, almost 20 million people were receiving a total of more than $4 billion in food stamps. But militancy had receded, replaced by 'poor people's' advocates, many of whom had cut their political teeth in civil rights and antiwar movements. So-called 'public interest' groups placed less emphasis on demonstrations and more emphasis on 'working within the system' (lobbying and litigation), in part, because they had to appeal to foundations that provided seed monies and operating funds. When ruling class fear of insurrection subsided, political system once again contracted. Soon afterwards, eligbility would be tightened and benefits would be begin to lag behind inflation, patterns that continue to present day. (See Frances Fox Piven and Richard Cloward on points made in this paragraph)
In sum, creation of food stamp program owes little (if anything) to left political activity, its dramatic expansion owes much (almost everything) to left political activity even if such activity was unable to force 'the system' to move beyond piecemeal concessions, bureaucratization, and state-enforced 'surveillance' associated with means-tested programs. Meanwhile, subsequent cutbacks in food stamps (which never amounted to more than about 1% of federal budget and always provided meager benefits) follow atrophy of strategy employed by groups like NWRO. Michael Hoover