The reason most major nations subsidize agriculture: food is the most strategic resource of all. Let there be a shortage of food, and the populace becomes very agitated. Hunger pushes people to do things they wouldn't normally do, like rise in rebellion. Humans can survive without a lot of things, but food is definitely not one of them.
All nations protect their internal agriculture, as a strategic initiative. A nation that depends on imported food becomes very vulnerable to an antagonist in times of conflict... if the antagonist can cut off those imports. The UK was very nearly put in that position during the Battle of the Atlantic - it was down to two weeks of food for its citizens at one point.
The only nation I could find that does not have agricultural subsidies is New Zealand. It is in a somewhat unique position, though. NZ has been a pacifist nation, partially out of desire, and partially out of happenstance - it is somewhat remote, doesn't dominate any major trade routes, or produce great amounts of strategic materials, so there is little reason for another nation to attack it, and little benefit to another country if they were to do so. Nor does NZ have a rapidly growing population, turning farms into subdivisions and industrial parks, and threatening their agricultural output.
Also, agriculture isn't like a factory. It can't be just stopped and started. A farm that has been fallow and unused for some time takes a few years to return to full productivity, whether that farm produced crops or livestock. So it is in the best interests of all nations to keep what farms they have productive.
Today, in the US, the bulk of agricultural subsidies consist of price supports. The government sets a price that it will buy agricultural goods, calculated to keep the farms producing, and private businesses have to pay more than that if they want the produce. In good years, the government buys very little. In rough years, they buy a lot.
Government purchased food often goes to famine stricken areas, such as parts of Africa. In one case when the cheese market tanked in the 80's, the government ended up buying a whole bunch of this perishable material on the price support program and giving it out to people on welfare... the Government Cheese meme.