Maybe a better question would be, what need was not being met by the religions already being practiced?
Islam spread in East Africa similar to the way the religion spread in Southeast Asia, that is, through trade followed by the emergence of the Islamic states that further facilitated the spread of the religion. This is why Muslim communities around the Indian Ocean are Sunni Shafiʿi. Islam was spread to Southeast Asia, East Africa, southern India, and various Indian Ocean islands (e.g. the Maldives, Sri Lanka, and Comoros) primarily by Shafiʿi merchants from places such as Hadramawt and Tihama in coastal Yemen. In the case of the Swahili Coast, the population was solidly Shafiʿi Muslim by the 14th century.
Though Indian Ocean trade networks existed for millennia and there were Muslim merchants and mosques present in East Africa since at least the 8th century, the first major conversions to Islam by the indigenous Swahili population took place only in about the 12th century. (This is the same century in which the Shafiʿi Ayyubids conquered Yemen and institutionalized Sunni Shafiʿism there.) As for your question about Islam fulfilling a "need [that] was not being met by the religions already being practiced," the following quote provides a good possible explanation.
"Many townspeople … operated in a wider world than the microcosm of the village, living in towns with other peoples, sailing from town to town along the coast, and trading with people from across the Indian Ocean. These people lived in a macrocosmic world inhabited by peoples speaking different languages, having different ancestors, and working in different occupations. In this world the beliefs of the microcosm were too parochial; what was needed were beliefs that were universal. And so townspeople began to adopt Islam, and in so doing they adopted a set of beliefs and a framework for action that were held in common by others in the town, by people in other towns, and by people from the whole Indian Ocean world."^1
In other words, Islam provided a way to bridge cultural, ethnic, and linguistic boundaries in the cosmopolitan coastal cities of medieval East Africa.
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