The hot springs in Manyara formed due to the penetration of water from highlands to the earth creaks, which are caused by the Tectonic force under the earth. Water reached the underlying rocks which are impermeable (means they don't allow water to penetrate further) as this underlying rocks are super heated due to the volcanic activities under earth, leads also water to get hot.
As in earth crust we have got high pressure leads the water to look for the way to get out. So after this water got the crack to move out, it moves outside/earth surface as a hot spring. The Sulfur: This which are found around the hot springs is formed due to the movement of water through variety type of underlying rocks washing/collecting different types of minerals, salts and sulfur which are accumulated in a site.(around hot springs)