This is a statement that is almost completely accurate. Britain's relations with non-African countries were decisive in the ways in which it controlled Africa.
There are instances in which these relations were not completely decisive. For example, the Boer War was not really caused by Britain's relations with European powers. Instead, it was about Britain trying to gain more control over people of European descent living in Africa.
Outside of that, however, relations with non-African countries were decisive. It was Britain's relationships with European countries (in the form of competition with those countries) that led it to try to take more formal control of various countries in Africa that it had only informally controlled before. Much later, it was Britain's relations with countries like the United States (with the US pressuring Britain for Cold War reasons) that pushed it to decolonize its African possessions.