Ehtiopia had traditionally had very good relations with Russia (the Ethiopian artillery that had outranged its Italian equilent in the Battle of Adwa 1896 had been bought from Russia), but after ww1, as Ras Tafari Makonnen, later Haile Selassie attempted to secure support, military missions and military expertise as well as a reliable source of arms, he mostly turned to smaller and neutral countries - partially because they had no amibition in East Africa, partially because turning to one Great Power risked angering the others.

Before the war, there were military missions from Switzerland, Sweden, Belgium and Turkey in Ethiopia. The Belgians trained and equipped the Kebur Zabagna, the Imperial Guard, the Swiss reformed and trained the Zabagna, the Addis Abeba Gendarmerie, the Turks worked with the mostly muslim feudal levies of Harar while the Swedes set up the Oletta Military Academy to train sons of the nobility in modern military tactics and warfare and worked with the northern feudal levies (Amharan and Tigrean christians) to teach modern military tactics, camouflage, entrenchment and so on.

Germany sold 12 37mm Pak35 AT guns, some 8 000 Mauser rifles, a small number of machine guns, a handful of MP 35 SMGs and a few million rounds for these weapons to Ethiopia before the French closed the port of Djibouti (connected to Addis Abeba by rail) for any weapon deliveries in accordance with the League of Nations weapons embargo on both sides in the conflict. The embargo struck Ethiopia much harder, as Italy had a domestic industry for the production of arms and ammunition while Ethiopia was completely dependent on foreign deliveries.

As to why the Germans supported the Ethiopians - at this time, Italy was allied with Britain and France in the so-called Stresa Front and Mussolini had in 1934 rushed 75 000 men up to the Brenner pass and the Austrian border when the Germans tried an Anschluss. Germany and Austria had to back down, not having the strength to face both Italy and France on land and the British on the sea.

The Germans were smarting from the Italian support for Austrian independence and saw a chance to strengthen the Ethiopians and cause the Italians to be occupied elsewhere, strengthening their strategic outlook.

However, Ethiopia fell quickly, and the Stresa front fell not soon after and Italy and Germany came much closer, to the extent that Mussolini did not object to Anschluss when it did happen in 1939.

Sources:

The Italian Invasion of Abyssinia 1935–36 by David Nicolle.

Haile Selassie's War by Anthony Mockler.