Simply the Admiral Graf Spee was in the South Atlantic to attack British shipping. I think I will leave narrative of the Battle of the River Plate to others, but I can give you some background as to why it was there in the first place.

Graf Spee was a Deutschland class “pocket battleship” and conceived and built for exactly this purpose. Optimised specifically to threaten British (but primarily in the context of the late 20s when she was designed, French shipping). Designed specifically to outgun French/British cruisers and outrun French/British Battleships. When she was launched only 3 ships in the West could on paper catch and out fight her (HMS Repulse, Renown and Hood). Their design had caused much furore on their launch. They remained just enough within the limits of the Treaty of Versailles to be acceptable yet beyond anything that the original draftees of the limitations had believed possible. Due to the diplomatic situation of the late 20s, they were specifically aimed at the French who had at that time no Battlecruisers that could catch them. Their appearance in part caused the French to withdraw from the Washington/London Treaty system in order build ships specifically to defeat them, causing Italy to follow in response. In their own small way these ships helped to collapse the interwar naval arms limitation process.

On the run up to war British assessments of German intentions were generally correct. The German surface fleet was not expected to concentrate in the north sea, they were expected to try and disperse and disrupt British maritime communication and supply. Attacks by surface raiders, such as the Graf Spee was essentially what interwar concerns had focussed on. The threat of submarines to trade, was thought to be manageable through convoys, new technology and faster transport ships. The threat of the surface raider was deemed paramount, guiding design, procurement, treaties and spending. Widespread successful interdiction of trade was viewed as the job for surface raiding cruisers, they had been a problem during the first world war for the British, the had also prosecuted the majority of the embargo of Germany by Britain, they also formed the focus of interwar plans for war against Japan. Defence against surface raiding cruisers, was also deemed to be more cruisers.

Britain itself felt specifically vulnerable to attacks on its sea lanes of communication. As the future First Sea Lord Admiral Pound stated: “Nothing would paralyse our supply system and seaborne trade so successfully as attack by surface raiders”. As a net importer of food and raw materials, Britain was acutely aware of its overseas trade. During the early stages of the war it was simultaneously importing large amounts of material from the Americas and continuing to try and maintain exports to places such as South America and the US. Britain’s long term planning in the 30s had been based in a large part on the concept of “the long war” and the treasury and economy being the “Fourth Arm of Defence”. In practice this meant that even during war, exports were still required to try and balance payments of import from abroad. South America was traditionally one of the largest markets for British exports outside the Sterling area, offsetting large imports of food (especially meat) from Argentina and Uruguay, oil from Venezuela and Central America. It wasn’t until after the fall of France and late 1940 that Britain abandoned plans to fight a long war based on sustainable armaments by exports and manageable debt. For example, Exports of Manufactured goods are nearly as high in 1940 as they are 1938. Exports to the US are 50% higher in 1940 than they are 1938. In 1940 Imports from Argentina are nearly as large as from India. All these slightly random stats illustrate that the Americas are essential to the British war economy both for imports and exports. Also don’t underestimate how important “business as usual” trading is to the value of the pound and the levels of inflation (and hence the cost of labour and arms) in the UK itself.

So….back to the Admiral Graf Spee. She and Deutschland had snuck into the South Atlantic prior to outbreak of hostilities in order to conduct long range commerce warfare and generally sow confusion and cause the British and French to commit forces in their hunt for them. A substantial collection of naval forces were assembled to conduct the search for them from a number of locations, including at points 5 carriers, 4 capital ships and 14 cruisers. Graf Spee sank 10 merchant ships off the coast of South America, totalling about 50,000 tons, before being located and damaged by 3 RN cruisers (Exeter, Achilles, Ajax) on the 13th December 1939. With heavier forces arriving the Graf Spee sought refuge in Montevideo, Uruguay. After much diplomatic wrangling and threats of internment the Captain shot himself after ordering his crew to scuttle the ship.

So the long story short is that the Admiral Graf Spee was in South American waters to conduct trade warfare against British shipping in exactly the way she was conceived and built for, conforming to much of the interwar concepts of how naval warfare would be conducted by, and against Britain.

Sources: I can be more specific if needs be. But the background economic and trade stuff comes primarily from the various works of George Peden, David Edgerton, and the Official History of the Second World War: A Statistical Digest for the imports/exports. The quote from Pound is from Churchill and Seapower by Christopher Bell p165. The general outline of River Plate can come from anywhere but I used Daniel Todman, Britian’s war and Eric Groves: The Royal Navy since 1815, p184.