A question that i can somewhat give insight! Yay!

This question, as another poster rightfully deduced, goes back to the problem the Liberian project faced since its inception by the American Colonization Society to convince freedmen to emigrate out of America. David Brion Davis's new book The problem of slavery in the age of emancipation goes in great detail over that.

Davis goes over the very strong reaction the black population mounted against colonization, and makes a strong case to the theory that a successful emigration/deportation of blacks from America would only strenghten the instituition of slavery in the United States, something that would be beneficial to the white slavers, not nearly as much to the black population.

Just imagine with me: what Davis argues is that, by removing the freedmen element from american society, the only social place blacks could occupy is the slave role. People like James Forten organized strong opposition to the ACS project of Liberia since its making, in 1817.

What is apparent throughout Davis's book is that Liberia is the brainchild of a project not made by blacks, that did not have the black population's welfare in mind, that was faced with combat from the free black population since its beginning. What makes Davis's book so marvelous is his extremely interesting breakdown on how the ACS masked this rather nefarious project with the face of good intentions, that goes like this: "you, black people are too corrupt to be part of american society because you'd been slaves, so, we'll send you back to africa to bring light and faith to that dark place, where you'll be a shining beacon of light and civilization".

That, as you imagine, did not bode well for the free black population, that understood the United States as their home, because of their generations-long roots, and thus did not embrace the project, as they sought success and integration in America, not outside of it.

Could you expand a little bit as to what you mean? Do you mean what led to the Liberian Civil War?

So, as others have said, it's unclear which failure you mean. If you're interested in the struggles of initial colonization, you might be interested in Tufuku Zuberi's book Swing Low, Sweet Chariot: The Mortality Cost of Colonizing Liberia in the Nineteenth-Century. He's a historical demographer at Penn, and I haven't read the book, I've only seen him talk (just a note, this book was published before he changed his name so on the book jacket I think it will say "Antonio McDaniel"). His argument is that black Americans suffered high moratality rates settling Liberia (and presumably Sierra Leone), similar not to native Africans but to other Americans and Europeans. Here's the blurb:

In the early nineteenth century, thousands of emancipated and freeborn blacks from the United States returned to Africa to colonize the area now known as Liberia. In this, the first systematic study of the demographic impact of this move on the migrants, Antonio McDaniel finds that the health of migrant populations depends on the adaptability of the individuals in the group, not on their race.

McDaniel compares the mortality rates of the emigrants to those of other migrants to tropical areas. He finds that, contrary to popular belief, black immigrants during this period died at unprecedented rates. Moreover, he shows that though the emigrant's mortality levels were exceptionally high, their mortality patterns were consistent with those of other populations.

McDaniel concludes that the greater the variance between the environment left and the environment entered, the higher the probability of contracting a new disease, and, in some cases, of death from these diseases. Additionally, a migrant's health can be affected by dietary changes, differences in local pathogens, inappropriate immunities, and increased risk of accidents due to unfamiliar surroundings.

As for the later political instability, if you believe the economic research by people like Daron Acemoğlu, these high mortality rates mean they were much more likely to set up "extractive institutions", which are worse in the long run for both economic growth and political stability. Similarly, there was an ethnic group of people in power (descendants of the American colonists) and groups excluded from political power (native blacks), which puts the country at risk for civil war, which broke out in the 90's and was devastating.

If you're asking why relatively so few freed American and British slaves ended up "back" in Africa, so it failed in its purpose, that's a different and complicated question, but the short answer is there appears to be no political will for such a solution in America. Likely, this is related to the fact that it would be economically devasting to regions that relied on slave labor to suddenly lose all their workers. As you can see from history, when the slaves were freed, laws and institutions were often set up to keep them more or less in place as pool of labor.