The word "manifest" means "obvious"; what is "manifest" is right in front of your eyes. The word "destiny" is easier: it is what is fated, what is meant to be. What is destined to happen will happen.
Therefore, Manifest Destiny stated that it was obvious that the Northern Europeans who constituted the free population of the United States were going to own the entire central North American continent, from the Atlantic to the Pacific oceans. This was not a debatable possibility: it was a certainty, at least to many people in power in the United States at that time.
If we look more closely at the use of the phrase in the 1840s, we find it was popularized by a newspaper editor, John O'Sullivan, who was a proponent for the US annexing Texas. He wrote in the summer of 1845 that it is
. . . our manifest destiny to overspread the continent allotted by Providence for the free development of our yearly multiplying millions.
Later that year, he used the phrase in a much more influential column about the Oregon territory border dispute, writing,
And that claim is by the right of our manifest destiny to overspread and to possess the whole of the continent which Providence has given us for the development of the great experiment of liberty and federated self-government entrusted to us.
His argument was that since the British had no interest in spreading democracy ("the great experiment of liberty"), they had no right to the Oregon territory.
The concept of Manifest Destiny expressed itself in the US's westward expansion in the 1840s into Texas and Oregon. An ideology that unquestioningly asserts that a nation has been ordained by God to control a territory is a powerful impetus to achieving the goal.
Clearly, the Native Americans, the British, and the Spanish had other ideas about this destiny, but the United States did prevail in obtaining most of the territory it sought.