Basically, it was a boom period, thanks to the war.

Remember, these countries were tied to the USA and parts of Europe (usually Great Britain) as markets for their exports of primary goods, such as metals, agricultural products or animal products. A lot of the labor force in European countries went to fight the war or were otherwise occupied in home production for the war effort, so less food and the like was produced domestically, leading to the need to import even more goods from abroad. The Germans, being unable to import basically anything themselves, used U-boats (primitive submarines) to systematically destroy many of the supplies that Great Britain and France did choose to import, leading to demand for even more products from South America.

Due to the rise in demand there was an increase exports and the value of these exports (so volume AND per unit value increased), meaning a huge windfall for many countries. (keep in mind, this is a generalization and does not apply to all cases at all times)

Then the war ended and there was a bust. Demand dropped, as did the per unit value of products, leading to a recession in many areas. The recessions were worse in some areas because in addition due to a lack of adequate planning for when the war inevitably ended and normalcy was restored. Many of these countries were already suffering systemic crises in terms of their dependence on one or two main basic, unrefined products anyway.

Also important to note: The recession isn't always immediate. In some places the slow down in demand took a while to truly impact South American economies.

[PS Brazil aided the allies militarily to some extend but it didn't really have a huge impact on the war and Brazilian beef honestly had a bigger impact than Brazilian arms]

For a great comparative analysis, country by country, try Tulio Halperin's Contemporary History of Latin America.