What is this mysterious pipe in my back yard?
This could be part of a larger yard drainage system. It could let water from the screened end of the pipe drain out into some buried drain field. It could also drain onto a deep buried french drain construction consisting of a sizeable hole that was filled with crushed rock or similar. 

The fact that the part of the pipe that has the screen is working its way out of the ground seems to imply that there is stress on the pipe from the end that is still buried. This could be due to settling at the buried end or the pipe was originally installed and buried by forcing it into the trench and then not covering it with enough soil. I bring this up because there is evidence that the filter screen end was buried at one time: