What you must understand is that mormonism didn't change for noble reasons. Every change was due to political or economical reasons.
Polygamy was dropped so Utah could become a state. The black ban was dropped because the US Government and NAACP threatened to remove mormons tax exempt status. But the biggest change came in the 1990s when Gordon B. Hinckley took over.
What I'm about to get into is quite a bit more recent than this sub usually allows, but I feel it's important to answering your question.
You see, the stereotype of the "really nice mormon" didn't start until Hinckley was in charge. Hinckley was a very different sort of leader than the church had before. He had a deep understanding of mormonisms image, and knew the power that things like television and the internet held.
He had several highly publicized television appearances most notably 60 Minutes and with Larry King. Something no other LDS leader had done before or since. This is also the time period when people of colour and minorities began to appear in mormon media and advertising, almost completely erasing the fact that just 20 years prior blacks were still banned from becoming members.
Serving missions became much more important. Several mormon filmmakers released several reasonably high profile budget films. This media blitz gave mormons a world presence that they never had before and created an image of a loving and intimate community, essentially exchanging the stereotype of creepy polygamists with a much more positive stereotype. All while the uglier parts of mormon history and doctrine were quietly swept under the rug.
And they might have more or less succeeded, if Hinckley hadn't died when he did. In the years since Hinckley's death and when Thomas S. Monson took over the LDS church has undergone a sort of regression in attitude. Things like the November 2015 Gay Ban, and dropping the BSA just a few weeks ago. Things which would have never happened 10 years ago.
In fact, given the changes in attitudes and policies it's very likely that the LDS church will continue to become more isolationist over the coming years, while the world continues to become more progressive and liberal. But this really isn't the sub for that kind speculation, and if you are still curious I recommend you seek out other places for that kind of discussion.
To make a very, very long story short: The changes that the church made for economic and political reasons led to a somewhat unintentional change in attitude. But this change in attitude, combined societal and technological advances resulted in a almost complete 180 in regards to mormons image and place in the world. However, beginning in 2008 mormonism has begun regressing back to its pre-hinckley attitudes, and will likely continue to do so into the near future.
I hope you found this helpful. =)