In the early stages of the fandom around 1990, it was all about art comics of animal characters until at one convention, someone showed up in a costume of their character (I think it was a lion in a loincloth) and impressed everyone around them. Soon people were making their own costumes that all honestly looked pretty awful but it was the start of the trend and many current fursuit makers. 

Naturally, the media jumped onto the costumes like hungry flies to a honey-covered stick and essentially came to the conclusion that fursuits were a necessity to be a furry. This eventually lead newcomers to believe they needed costumes too and were beginning to purchase them online around the 2000's. 

One costume maker listed the costumes under a heading that read "fursuit of happiness". This was meant to be a pun for "pursuit" but people took it literally-- which is quite easy to do-- as a suit made of fur. 

This is a little off topic but: Luxury Shag and a sort of mink-like fur is used and not usually real fur since that's a no-no between makers. Real fur is heavy and can get extremely hot and could lead to heat-stroke or other health complications. 

With the creation of fursuits around the same time as YouTube, these new "fursuit" makers began to share tips and tricks on the internet for others to learn from and better their own work. 

And with this constant sharing of knowledge, furs began to learn from each other and make fursuits look much better and even slip in other fancy things like magnetic eyebrows, screen faces (protogen), movable mouths, blinking eyes and so on.