Something that is impressive stands out because it is really good. Some of the most impressive college application essays are those that reflect on past events that shaped who the writer is today.
When you impress someone, you show that you are special or that what you did is praiseworthy. So something that's impressive does just that, like the impressive feat of running a marathon every week for a year. The earliest meaning of impressive was "capable of being easily impressed," from the Latin word impressus, "imprint or stamp." By around 1775, the meaning had changed to "making a strong impression on."