The answer to this can differ in some ways depending on what time period you are asking about. There have been different times in history that the US can be said to have “asserted itself” and the reasons why it could do so were not always completely the same.
There are at least some reasons that have been important every time the US has asserted itself. The main reason is that the United States is a very large country with many resources. A country can only assert itself in the world if it has enough resources to have military power. The large size of the US and the varied resources available in its landmass have allowed it to assert itself in the world. A second constant factor has been the geographic isolation of the United States from Europe. Europe was the greatest center of power in the world for a long time. By being isolated from Europe and protected from attack by the countries of that continent, the US was able to become more powerful. This allowed it to eventually assert itself on the world stage.
Other factors that allowed the US to assert itself on the world stage have changed over the years. At times, the US has been able to assert itself because its interests did not compete with those of important European powers. For example, the US set forth the Monroe Doctrine in 1823. At that time, the United Kingdom was much more powerful than the US, but the Monroe Doctrine was compatible with British interests so the US was allowed to assert this doctrine. At times, the US has been able to assert itself because other countries have been too weak to oppose it. This was particularly true after World War II. After that war, the US was able to dominate the Western World because all of the other countries had been devastated by the war. Finally, there have been times when the US’s own ideologies allowed it to assert itself on the world stage. One such instance was the time of the Spanish-American War. At that time, many Americans believed that it was right for the US to act in an imperialistic manner. Because so many Americans thought it was right to do so, the US was able to assert itself in the world.
We can see, then, that some factors allowing the US to assert itself have been constant over time while other factors have varied from time to time. If you want more detailed information about a specific time period, please specify the time period in another question.