Consider the two diagrams below (ignore the velocities),
Consider the situation in the right hand picture where we have a rod connected to the earth. Let us now say that the muon is created at the end of the rod, both frames (the earth and muon frame) most agree on this fact, since the laws of physics are the same in both frames. Then it should be obvious that according to the muon the rod is length contracted and thus it has a shorter distance to travail then an observer in the earth frame would say the muon had to travail. Now consider the left hand diagram where the rod is instead connected to the muon (and therefore at rest with respect to it). In this frame the rod will therefore be contracted to someone at rest in the earth frame compared to the muon frame (and thefore the observer in the earth would say the muon has a shorter distance to travil then someone in the muon frame would say). It should be clear that that these two arguments contradict one another. So why is the first analysis, where the length is contracted in the muon frame correct whilst the second is wrong.
(note that the rods are included simply for illustration)