What was manifest destiny? Why did Americans at the time believe it was just?

Manifest destiny was the belief that God wanted the United States to spread out over the North American continent, taking control of as much territory as possible.  Americans of the time believed in it because their country was strong compared to its neighbors and because this was a time in history when essentially everyone believed that might made right.
In the phrase “manifest destiny,” the word “manifest” means something like “obvious.”  The phrase, then, referred to America’s obvious destiny.  People believed that it was obvious that God wanted their country to succeed, grow, and spread.  This was part of what motivated the Mexican-American War and the westward expansion across the continent that snapped up Indian lands and pushed the natives onto reservations (or simply killed them).
People believed this because the US was clearly the strongest country on the continent.  Mexico was weak and constantly wracked with internal disorder.  Canada was not a country yet, just a colonial outpost of the British Empire that was no threat to the US.  The Indians were very few in number and lacked the technological sophistication to worry the Americans if the latter came west in any large numbers. 
In this situation, it made sense for people of the time to think that God favored them.  Across the world at that time, people believed that might made right.  They believed that a country was justified in taking land from other countries if it had the military power to do so.  They also believed that God decided everything about the world.  Therefore, they believed that he selected some countries to be strong because they deserved it and condemned others to be weak because there was something wrong with the people of those countries.
Because people of this time believed things like this, it is not surprising that they believed in manifest destiny.  Their country was clearly wealthier and stronger than those around it.  Therefore, it was clearly destined by God to spread at the expense of those other, weaker, peoples.
 
http://www.let.rug.nl/usa/essays/1801-1900/manifest-destiny/manifest-destiny---the-philosophy-that-created-a-nation.php

https://www.history.com/topics/westward-expansion/manifest-destiny