To fully understand the story you would have to know how the situation was in Ethiopia during the 30 or so years of war for independence.

The Eritrean war for independence began in the 1950s after the federation and later annexation of Eritrea by the Ethiopian monarchy, the early movements were dominated by Muslim Eritreans, but as Ethiopia s Grip on the region tightened the movement garnered more the support from Eritreans.

Now during the feudal monarchy rule of emperor Haile Selassie the Eritrean and several other ethnically organized oppositions in other parts of Ethiopia were seen as a side issue .

Emperor Haile Selassie was over thrown in 1974 after a popular revolution and replaced by the Derg (meaning the council), a committee formed by 108 low ranking army officers.

After taking power these officers initially chose an Eritrean born army General Aman Andom as their leader , but later a disagreement on how to handle the Eritrean issue led to his murder. After several other internal conflicts Mengistu Hailemariam Emerged as the leader of the Derg. The next steps the derg took to control the situation in Eritrean were so harsh that it turned the majority of Eritreans against Ethiopian rule.

At the same time opposition groups in other parts of Ethiopia were gaining power, mainly the TPLF( Tigray People's Liberation Front ) in the northern region of Tigray neighboring Eritrea had started an armed struggle to liberate the region of Tigray from Ethiopia.

The EPRP( Ethiopian People's Revolutionary Party ) which operated mainly in the capital and other major cities and was engaged in assassinations of Derg members and officials. This killings were known as 'white terror'. In response , the Derg waged the red terror , leading to the death of 100s of thousands of mostly young and educated Ethiopians.

Also in 1977 the eastern Ethiopian region was invaded by the Somalian government. So in its early years of rule the Derg was fighting wars in 3 fronts.

By the 1980s Ethiopia had won the war with Somalia in the east, the EPRP had lost most of its power and the country was under the dictatorship of Mengistu Hailemariam. He had absolute control over, which at the time was, the largest army in Africa.

So how did the largest army in Africa at the time, which had the support of the soviet union lose the war to the rebels.

1. The Eritrean liberation fighters had formed an unofficial coalition with the TPLF (the Tigriyan people in the recent census constituent 6% of Ethiopia's population). Both the Eritrean and Tigray rebels had support from the populations of Tigray and Eritrea.
2. The Ethiopian people were fed up with the oppressive Derg government. After the end of red terror , the government continued its political murders and also started a campaign of forcing young men to join the military.

Other armed liberation and opposition movements were also springing up in other parts of Ethiopia. Some forming coalitions with TPLF.

3. But the major cause for the government to lose the war was lack of competent leadership in the Military. As Mengistu consolidated power to himself , the army had adopted a leadership system called 'triangle command' in which inexperienced low ranking officers which were chosen based on their loyalty to Mengistu and were given the same power as experienced Generals. The old Chain of command was broken.

Mengistu also promoted his supporters to higher ranks , creating unnecessary political friction with the experienced generals.

The Army leadership was also plagued with corruption.

The experienced generals were forced to follow orders that came directly from Mengistu or his followers even if they didn't fit to what was happening in the ground. People who opposed any of the ideas proposed by Mengistu or his followers were executed. Officers who suggested finding a peaceful resolution were also executed.As the war continued , the army, which has been fighting continuous wars since derg took power, in regions which it had no support from the population, was losing morale. With the added confusion caused by the lack of coordination in the leadership the army started losing. and in some occasions commanders who lost battles were executed accused of conspiring with the rebels.

Mengistu focus on increasing the number of soldiers in the ground led to the decrease in quality of the army. In the final stages of the war he was planning to send university students to battle after only 3 months of training.

As one of the generals put it even though the army was numbered at 400,000 the number of qualified fighters they had wasn't more that 15,000.

The second most important cause was, a failed coup against Mengistu staged in Addis Ababa and Asmara(Eritrea) by high ranking military officers. These officers , who were veterans of multiple wars , were executed after a military trial, marking the beginning of the end for the Ethiopian Army.When Mengistu finally fled the country in 1991 the army collapsed , leaving the Eritrean and Tigray Fighters to control the capital with little opposition.

After Dergs fall TPLF took power and Eritrea had its referendum.

Source: An Amharic Book titled 'the reminiscences of colonel Mengistu Haliemariam', which contains interviews with several derg officials , Military leaders and also Mengistu Haliemariam himself.

If u want to know more on how the derg handled the situation in Eritrea check out this video from British channel ITV.

So in general the war wasn't between Eritrea and Ethiopia , it was between the Eritrean rebels and the Ethiopian Government which was also facing opposition from Ethiopians too.