I'm not sure if you're still looking for an answer to this, but it's a complicated question with multiple parts.

The True Whig Party was actually a de facto one party state for most of its existence, not a de jure one. That means that there wasn't a ban on multiple parties, but the one party so thoroughly dominated Liberian politics that there was almost no point in being part of a secondary party. This briefly changed during the regime of the last True Whig president, William Tolbert, when the Progressive People's Party became a viable opposition party.

The True Whig Party dominated Liberia for so long because it was the party that represented the interests of the Americo-Liberians, the descendants of freed slaves from the United States who immigrated to what would become Liberia.

When they arrived in the area, the local people were divided across many tribal groups that spoke different languages, and had different chieftains and religious customs, many of which involved human sacrifice and cannibalism (this infamously made a comeback during the Liberian Civil War). Anything resembling a modern state didn't exist. The Americo-Liberians originally started Liberia as a nation for them alone, in the territory they had settled in. Over time they annexed the surrounding land and the tribes who inhabited that land, but incorporating the tribes into the Liberian nation and government was very slow going and met with resistance on both sides. The Americo-Liberians mostly wanted to retain their privileges as a societal elite, while the natives didn't understand the central Liberian government's western style democratic system. So for a long time the Americo-Liberians just used the natives as a source of tax revenue and otherwise let them do their own thing, only interacting with them through their local chiefs. Tribals were basically not included in the electoral process.

To modernize the country, the tribal groups had to be incorporated more and more into the system, which meant that they were getting more educated about the Liberian government and participating in the national Liberian organizations. But the system was still very unequal. Tribals had lower ranking positions and more grunt labor oriented jobs, while Americo-Liberians dominated administrative positions. This is what eventually caused the True Whig Party's downfall, because one organization that worked like this was the military. The officer's corp was all Americo-Liberian, by requirement, but the foot soldiers and NCO's were filling up with members of the tribal groups, who were also getting increasingly unhappy with their second class citizen status. So not surprisingly, eventually a group of those tribal soldiers and NCO's got together and overthrew the government in a coup. Unfortunately the guy who they put in charge, Samuel Doe, turned out to be worse than what came before, which led to the horrific Liberian Civil War.

But the moment that coup happened, the True Whig Party's dominance was over. From that point on, tribal Liberians were a part of all future elections in the country, and since the Americo-Liberians were severely outnumbered, their chief party, the True Whig Party, became a minor party.