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.

The spirit of manifest destiny showed itself in the American expansionism of the 1840s because manifest destiny was the idea that the United States was destined by God to expand across the entire continent of North America.

The idea of manifest destiny held that the United States was a superior nation chosen by God to be great.  Believers in the idea felt that the United States had the best system of government in the world.  They felt that Americans were racially superior to the Native Americans and to the Spanish in Mexico.  They felt that American Protestantism was superior to the Roman Catholic faith practiced by the Mexicans as well.

These attitudes led to the American expansionism of the 1840s.  They led to the idea that the US deserved to take much of the territory of Mexico in the Mexican-American War.  They also led to the idea that the US deserved the have things like the Oregon Territory. 

Manifest destiny was an expansionist idea that was used to justify the American expansionism that occurred in the decade of the 1840s.