Greg Laden, who has been to this region many time, has some personal observation about the relationship between religion and political culture there:
1) The Missionary Influence is important. Tanzania is far less missionized than the other countries in recent decades, and Rwanda is a bid different, but there is a pervasive religious factor that is Christian, and subsumes the thinking of a goodly part of the population. It is not the case that everyone in the region is a fundamentalist Christian. Many people in East Africa and adjoining regions are Muslim (a Majority if Kenyans, I believe), and many are relatively indifferent to religion. But it is a factor.
2) Many of the human-rights related issues that we see from a Western perspective as being important are more developed in East Africa than a lot of westerners might expect. Women have been in positions of power in the government for years, and there are open movements to fight violence against women and other patriarchal features of human societies. Although there is just as much "inter tribal" (= inter ethnic) crapola in East Africa as there is in Europe or the US, there is also a much better and more advanced understanding of racial issues. In short, there is a progressive attitude regarding human rights. Well, some human rights.
3) In some subcultures in the regions, homosexuality is normal and integrated into day to day life, or ignored. It is not a big deal.
4) At the same time, where the Christian missionized influence is strong, homosexuality is considered to be one of the great wrongs against god. Homosexuality is seen as an abomination, and "abomination" isn't just some quaint word out of the Bible.
Uganda in particular is run by a government that is nearly explicitly Christian and fundamentalist, and there are many ties between the Ugandan government and the church. Many of the internal civil conflicts that have occurred over the last decades or that are going on now are explicitly religious.
Thanks to Eric O for providing a link to this
Religious theists -- and even some atheists -- object when so-called "new" atheist critics allege that "religion poisons everything." Well, even if religion generally does not poison everything, some religions certainly can inject poison into far too many things where they are introduced. This is just one depressing example of such a phenomenon and it's certainly a good reason to object to religions like Christianity. We've got enough problems in America without our petty religious bigotries being spread abroad.