It's a very popular fable, and very truthy; but I have never seen any evidence for it.
Farmers in Ethiopia in the mid 1980s were already out of business. The main reason was war; the struggle for Eritrean independence in particular, and the other usual tribal conflicts that are the inevitable consequence of borders drawn by European power brokers in the 19th Century with no regard for existing tribal boundaries meant that most of East Africa was a basket case - AK47s and ammunition were cheap, millions of young men were unable to work, and stealing stuff at gunpoint is easier and more fun than growing crops (that are going to get stolen at gunpoint anyway).
The region was almost able to feed itself in good years; but the drought was the last straw, and millions starved. At that point, sending free food wasn't going to do any harm to anyone; and it certainly saved a lot of lives.
Some 'free food' is still being sent to East Africa; the Somali refugees who have fled across the border to Ethiopia and Kenya to escape the latest famine are getting it, which takes some of the burden of feeding them off their hosts.
The population of Ethiopia today is more than twice what it was at the start of the 'Live Aid' famines of the 1980s; the same drought conditions have struck again in the last couple of years, with twice as many mouths to feed - and yet the disaster is missing, limited to Somalia.
In the 80s, Kenya had no disaster, while Ethiopia and Somalia starved. In the past five years, Kenya and Ethiopia had no disaster, while Somalia starved. These countries share the same weather pattern, and the borders are very porous, so food can be traded where needed; The population today is more than twice what it was; and the weather is just as bad. The difference in outcomes is not due to farming, or food prices, or climate, or population. The clear and obvious difference is rule of law. A farmer who knows that his crops will make it to market without being stolen, will grow those crops, sell them, and buy whatever he needs for his family with the proceeds.
That a few million refugees down the road are getting 'free food' from the UN has no significant negative effect on him; If he has food to sell, he can sell it to the UNHCR, or to the local market, or for export. Quite likely he is growing oilseed or coffee anyway; if food is cheap locally for any reason, that actually helps him, because he spends less of what he sells his coffee or for on food, and has the chance of having some money left over to buy other things he needs.
In recent years, flowers for export have become a major crop in Ethiopia; subsistence farming is no longer a way of life for many Ethiopians, and as a result, famine is no longer a way of life either. Growing flowers is a better way to feed your family than growing wheat or sorghum, if you can sell the flowers for enough cash to buy more grain than you could have grown, and still have money left over. Trying to compete with cheap grain from American and European agribusinesses isn't a good plan - and Ethiopian farmers ain't dumb.
If we assume that Ethiopian farmers are stupid; that people in refugee camps are in a position to grow food if only there wasn't all that free rice being handed out; that the situation in East Africa is simple; and that the people living there are little children who need our guidance, and some tough love, then the parable of the free rice seems almost plausible.
Sadly, none of those assumptions are reasonable; and oddly, nobody ever seems to have any evidence of a poor farmer who lost his livelihood to the dreaded foreign aid of free food.