The answer is kind of complicated. Hopefully I can give a solid enough answer but it might require some back story. Mormonism started in New York but was forced to relocate its base of operations several times because of persecution. Eventually they ended up in what is now know as Utah(was currently a part of the Republic of Mexico at the time but shortly afterwards became a US territory). Mormons began openly practicing polygamy around 1852 however polygamy wasn't practiced by most members. In some towns the percentage seems to be recorded at around 5%. So when the U.S. government started to push forward anti-polygamy laws the Mormons were very defiant. They argued religious freedom and therefore the laws weren't enforced in Utah.
Eventually with the Edmunds Act in 1882 the U.S. government got stern enough with the laws and started enforcing them in Utah. Polygamist were fined and or imprisoned. During the next seven years several more laws were pushed forward that disincorporated the Mormon church and made several of its leaders go into hiding in order to avoid arrest and jail time. Finally in 1889 the Mormon Church and therefore Utah agreed to do away with polygamy in order to avoid the Cullom-Struble Bill.
So this is where we get to your question. In September of 1890 the President of the Mormon Church issued a statement that said "I publicly declare that my advice to the Latter-day Saints is to refrain from contracting any marriages forbidden by the law of the land." After this the members of the church were expected to obey the law. However men interpreted this in several different ways. Some believed this didn't apply to excising marriages and therefore they could stay married to all their wives. Some left their wives(I'm assuming they just choose one but I'm unsure how these households turned out). Some men continued to live with and support all their wives even if the law didn't acknowledge they were still married. Some even found loopholes in the law(having one wife in the US and one in Canada). By 1904 the Mormon Church started to excommunicate members who were still in Polygamous marriages.
Hopefully this answers your questions. I'm going to bed now but if there are need to provide sources, answer questions, or correct grammatical errors then I'll gladly do it when I wake.
Edit: Please read some of the replies to my post. There are some very well informed people that make some follow ups and corrections to what I have said.
Edit: I understand that some people have some issues with some of the statements I've made. I tried to leave some of my facts ambiguous because I knew there are some exceptions. Yes Polygamy did still happen long after the Manifests and announcements. However; for the most part I felt like I answered OP's question correctly. Some cohabitation continued, some marriages continued(though not legally), and some men went to a single wife.
Ex post facto laws, or laws that make something that already happened a crime (or worse crime), are explicitly prohibited by the U.S. Constitution (Article I, Section 9 and Article I, Section 10). While there are some legal exceptions, it's more likely that they were left alone and dealt with by social pressure. I don't know the specifics of this case, but in general the law won't affect things that were previously legal.
Why was polygamy important in Mormon doctrine to begin with?
There's one thing I haven't seen mentioned here, so I'll mention it.
Leading up to the 1890 "Official Declaration", there had been a number of laws passed by the federal government with the aim of stomping out polygamy. Each time, the Church raised the defence of "Hey, First Amendment, religious freedom!"
But in the Supreme Court of the United States issued two rulings that basically said "Sorry, we're saying you don't have the freedom to do this." One was Reynolds v. United States, 98 U.S. (8 Otto.) 145 (1878). George Reynolds had been arrested and charged with bigamy (the Morrill Anti-Bigamy Act of 1862). His defence was "Hey, my religion commanded me to do this, so I'm protected by the First Amendment!" SCOTUS said "Nuh-uh, 'freedom of religion' is not an acceptable defence for breaking a criminal law":
In our opinion, the statute immediately under consideration is within the legislative power of Congress. It is constitutional and valid as prescribing a rule of action for all those residing in the Territories, and in places over which the United States have exclusive control. This being so, the only question which remains is, whether those who make polygamy a part of their religion are excepted from the operation of the statute. If they are, then those who do not make polygamy a part of their religious belief may be found guilty and punished, while those who do, must be acquitted and go free. This would be introducing a new element into criminal law. Laws are made for the government of actions, and while they cannot interfere with mere religious belief and opinions, they may with practices. Suppose one believed that human sacrifices were a necessary part of religious worship, would it be seriously contended that the civil government under which he lived could not interfere to prevent a sacrifice? Or if a wife religiously believed it was her duty to burn herself upon the funeral pile of her dead husband, would it be beyond the power of the civil government to prevent her carrying her belief into practice?
So here, as a law of the organization of society under the exclusive dominion of the United States, it is provided that plural marriages shall not be allowed. Can a man excuse his practices to the contrary because of his religious belief? [98 U.S. 145, 167] To permit this would be to make the professed doctrines of religious belief superior to the law of the land, and in effect to permit every citizen to become a law unto himself. Government could exist only in name under such circumstances.
The second key ruling was The Late Corporation of the Church of Jesus Christ of Latter-Day Saints v. United States, 136 U.S. 1 (1890), in response to the Edmunds-Tucker Act of 1887 already mentioned. That Act basically called for the disincorporation of the Church, and the seizure of all its assets. Not surprisingly, this SCOTUS ruling, which also makes much reference to the Morrill Act, includes much of the same language that the 1878 ruling included:
One pretense for this obstinate course is that their belief in the practice of polygamy, or in the right to indulge in it, is a religious belief, and therefore under the protection of the constitutional guaranty of religious freedom. This is altogether a sophistical plea. No doubt the Thugs of Indiaimagined that their belief in the right of assassination was a religious belief; but their thinking so did not make it so. The practice of suttee by the Hindu widows may have sprung from a supposed religious conviction. The offering of human sacri- [136 U.S. 1, 50] fices by our own ancestors in Britain was no doubt sanctioned by an equally conscientious impulse. But no one, on that account, would hesitate to brand these practices, now, as crimes against society, and obnoxious to condemnation and punishment by the civil authority. The state has a perfect right to prohibit polygamy, and all other open offenses against the enlightened sentiment of mankind, notwithstanding the pretense of religious conviction by which they may be advocated and practiced.
Anyhow. tl;dr Only after the Church had exhausted all avenues for appeal did then-Church President Wilford Woodruff basically said "OK, we're done here," and issued the 1890 proclamation renouncing polygamy.
This is from a LDS church article on the history of plural marriage:
After the U.S. Supreme Court found the anti-polygamy laws to be constitutional in 1879, federal officials began prosecuting polygamous husbands and wives during the 1880s. Believing these laws to be unjust, Latter-day Saints engaged in civil disobedience by continuing to practice plural marriage and by attempting to avoid arrest. When convicted, they paid fines and submitted to jail time. To help their husbands avoid prosecution, plural wives often separated into different households or went into hiding under assumed names, particularly when pregnant or after giving birth.
(https://www.lds.org/topics/plural-marriage-and-families-in-early-utah?lang=eng)
Some more reading on the persecution years of 1877 to 1887
Another on What is called Years of Endurance
Why did/does the federal government care so much about people's marriages? Is it just property/monetary control over the population they seek? What's the point? All I've seen was to protect white southern women from being whisked away to Utah. Why the religious persecution?