I can't speak for the others you mentioned, but Ethiopia managed to fall into (comparatively) fortunate circumstances in which competing European powers all had different aims in the Horn of Africa and thus were willing to support the Ethiopians to undermine the others.

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Backing off a few decades, 19th-Century Ethiopia was ravaged by some pretty constant conflict due to problems both at home and abroad. Traditional feudal hierarchies were a major part of governance, and the power of the Emperor was often at odds with that of local rulers. European intervention in Ethiopia really started in earnest under Tewodros II, who had requested the support of Great Britain to help crush revolts and encroaching neighbors as he tried to modernize the country. When he received a reply of "no" three years after his request was sent, Tewodros imprisoned the British messengers, prompting an intervention that ultimately resulted in Tewodros' death. The next Emperor to come along was himself outmaneuvered within a few years by the future Yohannes IV, who would continue Tewodros' general trend of trying to fend off invaders (Egypt and Madhist Sudan, in this case) while crushing rebels and consolidating power. Though more successful than his predecessors, he spent most of his reign fighting enemies foreign and domestic. Among the latter category was King Menelik of Shewa, who had been skillfully maneuvering for years to build his power base. Among his more local attempts to gain support, he also reached out to the French and Italians in their newly established colonies in Eritrea and Djibouti for modern weaponry. Ultimately, after years of preparations, he was well-positioned to seize power when Yohannes was finally killed in battle against the Madhists in 1889.

Almost as quickly as he moved to usurp the throne, Menelik also made a treaty with Italy. The Treaty of Wuchale nominally was supposed to clearly define the border between Italian Eritrea and Menelik's Ethiopian Empire. Unfortunately for Menelik, the Amharic and Italian versions of the treaty featured a slight discrepancy in a clause in one article. In short, the Italian version claimed Italy had the right to Ethiopian foreign affairs, effectively making them a protectorate, while the wording of the Amharic version of the treaty did not make this the case. Once this discrepancy became known to Menelik, he disowned the treaty and tensions started to boil over.

Here's where the competing foreign powers come in to play. France, Britain, and Italy all had their eyes on Ethiopia for various reasons. In addition, politics back in Europe were spilling over in the Horn of Africa, as the Franco-Russian alliance was very much at odds with the German-Italian-Austrian Triple Alliance. While Russia didn't have any colonies in the region, the Tsar styled himself as a defender of the Orthodox faith, providing the same rationale for support and intervention that was used to justify intervention in the Balkans. So the French continued to supply materiel support and the Russians joined in, even providing something like 30,000 Berdan rifles, artillery pieces, and military advisers to Menelik. The British and other members of the Triple Alliance were more supportive of Italy, but weren't willing to provide active support for such a colonial venture (especially with the British aims in the region at odds with the Italians).

Ultimately war did finally break out in 1894, with the Italians sending an expeditionary force from Eritrea. Unfortunately for the Italians, Menelik was able to field a massive army from Ethiopia's constituent kingdoms - Menelik's contingent at Adwa alone outnumbered the entire Italian force - and was fighting on his home turf while the Italians were fighting in unfamiliar terrain and beset by economic troubles at home. The Italians made initial gains, but were ultimately crushed at Adwa and routed back to Eritrea. Following the crushing defeat, the Italians negotiated peace with Menelik, with the new treaty clearly stating Ethiopia's independence and expanding and delineating Ethiopia's borders to effectively what they are to this day.