Do you want the LDS version or the true version? I’ll give you the true version. Joseph Smith had a problem with marriage fidelity. He had affairs behind his wife’s back. But he knew this couldn’t fly for long, him being the prophet and all, so he had a “revelation” that God was instituting plural/eternal (celestial) marriage. This revelation later became known as Section 132 of the Doctrine and Covenants, which is canon scripture in Mormonism. Joseph kept this revelation secret for some time and started marrying other women, a lot of other women…somewhere around 50 before he died, including women already married to other men and at least one 14 year-old, probably more.
Joseph tried to keep this secret from his first wife, Emma, because he knew she would most definitely not approve. She did eventually find out and was furious. She confronted him about it and demanded that he stop. So instead of doing the right thing, Joseph doubled down and had another revelation from God that Emma must accept his additional wives or face destruction. Well, apparently Emma didn’t like the sound of that, so she reluctantly capitulated. This wasn’t the only time Joseph used a threatening “revelation” to get someone to bend to his will. He did the same thing to some of the other women he wanted to marry.
To answer your first question, D&C Sect 132 is the reason eternal marriage is necessary to reach the celestial kingdom. It’s all nonsense of course, but in Mormondom it’s taken very seriously, although not fully–but that’s another topic.
As to your second question, no, in Mormondom it’s not enough for a woman to stay single and live for Jesus. The only way to the celestial kingdom is to marry in the temple. Mormons claim that if you don’t you will have the chance to do so in the afterlife, but if a single LDS woman told her family or friends that she wanted to stay single and live for Jesus, they would look at her like she’s crazy.
In the beginning, polygamy was kept to the inner circles of Mormon leadership. But when Brigham Young took over and moved the religion to Utah, he went full-on polygamy and took it to a whole other level. Being so far away from the authority of the U.S. government allowed him to do whatever he wanted, but even that did not last.