The idea of manifest destiny helped give rise to later American imperialism because it provided a justification for imperialism. It gave Americans a way to argue that their imperialism was actually a good thing for those who were being conquered. It gave them a way to claim that they were not just acting selfishly.
The idea of manifest destiny was that God had destined the US to expand over a large area of land. God had done this because the US was a superior country. It had a superior form of government. It had a superior religion (not just Christian, but Protestant in particular). It had a superior culture. Therefore, it deserved to expand.
The same ideas helped to justify later imperialism. It was good for the US to take the Philippines, for example, because Americans were superior to Filipinos. They were Protestant where the Filipinos were Catholic. They had democracy where the Filipinos did not. For these and other reasons, conquering the Philippines would actually be helping the Filipinos.
In this way, the idea of manifest destiny provided a justification for later acts of imperialism.