Simone:
I'm going to take a slightly different tack on the 'why is it important' question. In some ways, the most important reason is because you'd like to get an 'A' in the class, and one way or another, identifying the stages is likely to be on exams and quizzes.
In thinking about the stages, it helps to think about history. They were named when the best way to learn about meiosis was to look through a microscope and see what you could see. So the stages correspond to easily identified changes in the appearance of cells using light microscopy.
Now the reason that there were visible differences is because there was stuff going on--stuff that was critical to the accomplishment of the goals of meiosis. Meiosis exists because for sexual reproduction, we need each parent to contribute EXACTLY HALF of their genetic material to offspring. That's an incredibly challenging task... but if you think about what MUST happen, you can pretty much design meiosis from scratch, and predict what things will look like.
For example: if you're going to be parceling the genetic material out into two compartments, you're going to have to drag the chromosomes around. In order to drag them, you must have free access to them... but they're trapped in the nucleus! So you're going to have to get the nucleus out of the way... which means at some point in your microscope, you're going to see it disappear.
Since you're going to need machinery for dragging, that's going to have to get built, and you'll see that (spindle)
In order to get 'one of everything' into each daughter cell, the only way to reliably do that is to first pair things up, then pull the members of the pair in opposite directions. So at some point you're going to see the homologs pair (meiosis I) or the sister chromatids pair (they are actually 'born paired' because they never separate after replication). This would work best if all the pairs 'lined up' in a uniform way... and that's exactly what you're going to see in your microscope--all the pairs in a row. Then you're going to have to drag them apart, so you'll see that to.
tl;dr: the stages correspond to critical, observable events that are part of meiosis serving its purpose