Many Republicans (particularly those shouting the loudest and saying that Obama ruined America for 8 years) of them disagree with many or most of the things he did, hurting the country's military, economy, and taking us in the wrong direction, socially.
Let's look at some things he did which some conservative Republicans may object to:
Passed the Affordable Care Act ("Obamacare"). This was seen as subsidizing health care for poor people. Insurance rates for some middle class citizens and upper class saw increases to help pay for those who pay little to nothing for their health insurance. Many Republicans take the stance that individuals should prosper or fail by their own accord. If people can't afford basic health insurance, then they should work harder, get better educated, and find a better job, not ask the government for handouts.
Paris Accord on climate change. A number of them do not think that man-made climate change is real and restrictions agreed upon at the accord means delaying progress, hurting the economy and businesses that are demonized, and putting money into the pockets of "green" businesses like solar energy.
Iran nuclear deal. Many feel that Iran has not and will not live up to any kind of negotiated deal. The most extreme feel that only the military might will make them see reason and stop any and all development of nuclear power.
Legalized same-sex marriage. If you view homosexuality as a mortal sin, then making gay marriage legal is equivalent to saying that it's okay to be evil, do sinful things, and the government will not just look the other way, but condone it. Religious fundamentalists see this as the whole country literally going to hell.
Repealed "Don't Ask, Don't Tell". Tied in with the previous point. Some think that the armed forces are the country's military and represent its values; allowing gay men to serve is an abomination and an affront to God. There's also an argument being made that it compromises soldier safety (not coincidentally, many of them object to women serving for the same reason). You don't want soldiers to fall in love with another in their same company and risking the lives of everyone else to save one person. For the Star Trek nerds, re-watch the DS9 episode, Change of Heart to see their worst fears realized: on an away mission, Worf aborts the mission to save his wife, leading to the death of another operative.
Tightened EPA restrictions on coal plants. Many object to tigher regulatory standards, particularly when it hurts businesses. Many of them will argue that by forcing them to adhere to these standards, it means that some coal miners will be laid off, some other workers will, too, and for a nebulous reason like "climate change", which they don't think adversely affects Americans as much as environmentalists claim.
Nominated Sonia Sotomayor and Elena Kagan. He nominated two liberal women to the Supreme Court. If we had a Republican president, we likely would have seen two conservative men instead.
Boosted budget for the FDA. Like the EPA, the FDA imposes tigher restrictions on businesses for health and safety standards. This means those businesses are less profitable when they have to spend money to protect their workers and consumers.
Reduced defense spending. Many Republicans are opposed to reducing the military at all, and some would like to see it increased. They argue that a smaller defense either puts American lives at risk or means that fewer are able to be deployed to fight terrorism abroad.
Renewable technology investments. At the same time that Obama made it harder for coal industries, he also gave money to solar energy. Many think this is absolutely corrupt and kowtowing to liberal environmentalists.
Healthy Hunger-Free Kids Act: Michele Obama is largely the face of this, but a lot of conservative parents take objection to the government invading in their lives and telling them how to raise their kids.
Hate Crimes Prevention Act in 2009. Many argue a crime is a crime and we shouldn't punish people for thoughts in their heads. Legislating that crimes motivated by racism, sexism, or homophobia is overreaching and unecessary.
Children's Health Insurance Authorization Act. This allows kids to stay insured longer, but hurts the health insurance industry. Many Republicans think that it's best if we have a very laissez faire attitude when it comes to the economy and interfere as little as possible. Forcing insurance companies to pay more either means hurting businesses directly, or more likely, they will raise premiums on other payers.
tl;dr: Obama hurt America's economy and is literally sending us to Hell. They want a Republican back in the White House to reverse course.