The simultaneity of events can get complicated when talking about relativity. Einstein tells us that due to relativity, an event observed by two different reference frames is not observed as simulations if one frame is moving.
But what does that mean? It means, for example, that if you are running very fast (near the speed of light) while holding a rod with two identical flashing strobe lights on the tips, you will always observe the lights flashing at the same time in your own frame of reference. However, if you run passed one of your friends, they will not see the lights flashing at the same instant due to relativity.
So in the speeding train example, we have a similar situation. However, the roles are reversed. What we call the stationary observer (on the platform) sees simultaneous flashes at either end. When the speeding train goes by, the moving train observer does NOT see the flashes as simultaneous!
Other weird things will also happen. Not only will the you and your friend view time differently, you will also view the length of the rod differently! Your friend will think the rod is actually a shorter distance than you think! Now things are getting really wacky and cool!