SHENZHEN, China—The top player in Africa’s fast-growing smartphone market isn’t Apple Inc. or Samsung Electronics Co. It is Transsion Holdings Ltd., an obscure Chinese manufacturer that won customers by offering handsets with features targeted to local markets.

Transsion—whose products are sold under the Tecno, itel and Infinix brand names—developed handsets with two SIM-card slots after research showed people were carrying additional cards to avoid making out-of-network calls to save money. It also optimized its cameras to...