In the Last of Us the infected are just infested by a parasitic fungus known as Cordyceps, which is a real fungus with the ability to take over the body of an insect or spider and other critters like that. They force them to walk to their colony and then break stems through their bodies to release spores to infect the rest of the colony with the parasite. In The Last of Us it evolved to affect humans. So basically once you get infected you turn into a Runner, you look like a human but you just have no control over your body though you can still see, feel, and think. Over time the stems start to break through your skin and over a few years it will completely take you over, you will die, and it will just control your body. Once they "turn into a mushroom" like you said, they begin using echolocation to see hence the name "Clicker". 

Runner - Still alive, no control over their body
Stalker - Still alive, no control over their body, but the fungus starts to be able to use tactics to find its foes
Clicker - Dead, uses echolocation to see
Bloater - Dead, uses echolocation to see, is very hard to kill as the fungus has grown across its whole body to form plates that act as armor. 

Hope this helps you to understand why they look so odd.