In short, supposedly the musical is a parody about LDS missionaries who go to Africa on a mission, but are totally unprepared for the real world problems they encounter such as AIDS, starvation, etc. Mr. Otterson defends the church by telling of the humanitarian aid that the church has given to Africa, and also of the devotion of the LDS missionaries to the cultures in which they have served.
Personally, as someone who has served a mission for our church in a foreign country, there is one more aspect to our service that he neglected to mention. We do have missionaries who are called as "welfare missionaries", who go throughout the world helping nations to farm, and dig wells, and get vaccinations etc. But the bulk of us are there to share the gospel of Jesus Christ. While we may not always be able to help them with the physical trials they face, what we offer them is a sort of spiritual vaccine for sorrow, death, and sin. No, we may not fill their bellies with food, but our purpose is to fill their hearts with the spirit of God. And what missionaries have found is that those who have been humbled by natural disasters, or plague, or poverty, or famine, have been those who seek God most earnestly.
I don't claim to know everything about the culture in which I served, but I have taken in to my soul the parts about which I have learned. I have loved the people, and I have strived to honestly serve God. If that is something in which others find humor, then maybe they have missed the point.