It may help if you could be a little more specific; even then it's unlikely you'll get any kind of definative anwer.
Assuming you're asking about the decline of the Zweihander and the rise of the archetypal cavalry sabre the answer is two fold.
There is no way of knowing what caused the decline of the Zweihander with any real certainty; the best answer I have seen is that the rise of combined arms formations, such as the Tercio, and the growing efficacy of Firearms in the mid 16th century, the same time the Zweihander was abandoned, caused the decline but remember that correlation =/= causation.
Straight, double edged swords did not fall out of use until cavalry did. As evidenced by existing equipment and Contemporary depictions Cuirassiers of several nations, France and Prussia for instance, we're issued straight, double edged swords.
That's a very vague question. Do you refer to what happen between the swords of the 'middle ages', which are double-edged and cruciform, and the cavalry sabres of the 18th century or so, which are more often curved and single edged?
To put it simply, a large number of changes, over several hundreds of years, necessitated different forms of swords to be developed and used, and no single reason and event really dictated the change.
In other words, the change didn't happen overnight.
I will explain further if this is indeed what you are asking for.
Everyone is answering one side of the question but ignoring a big part of the other - there were plenty of single edged swords in the middle ages, and these were incredibly popular - so mch so that most of the remaining ones are in incredibly poor condition because of overuse. These included swords like falchions as well as smaller bladed weapons like the seax. There are also numerous examples of single-edged cruciform swords, as well.
That opens up a whole new question about curvature of a sword and why it was more popular in some cultures and some eras over others. As far as my own research goes, there is a lot of conflicting information to wade through and I have not seen any satisfactory thesis on the merits of curvature in swords. As curved swords existed before, during, and after the middle ages, I think the long-held assumption that curved swords are a "better" evolution is simply inaccurate and ignores many of the advantages of straight swords as well.