There are a number of reasons. First of all, American businesses looked for new markets abroad, and so some kind of imperial policy was thought necessary to achieve this goal. European colonial powers such as Great Britain and France had pursued such a policy for many years with much success; and as the American economy was rapidly becoming the world's largest, there was a certain logic to the United States emulating their example.
With growing economic power went growing political and military power. The United States was now recognized as a major player in international politics, and imperialism was a natural outcome of this. European countries had enhanced their standing on the world stage by the acquisition, development, and exploitation of their colonies, and it was thought that the United States might also do this. No longer was it enough for the United States to see itself as a beacon of liberty or the land of opportunity; it must also be respected in the old-fashioned way: as a colonial power.
Nevertheless, American imperialism was of a somewhat different character from that pursued by the Europeans. For one thing, strategic influence was considered more important than the simple acquisition of territory. Americans understood the value of soft power in addition to more assertive expressions of military might, or as Theodore Roosevelt famously put it, to "speak softly and carry a big stick."
Whatever the consequences of imperial policy may have been, there's no doubt that many of the architects of this policy genuinely believed that the United States was spreading the principles of American liberty abroad, a liberty that was presented favorably in comparison with European colonialism and its effects. American imperialism was often presented in an emancipatory guise. We can see this, for instance, in the assistance given by the US government to Cuban nationalists in driving out the Spanish and also in the occupation of the Philippines by the Americans after they had defeated Spain in the Spanish–American War. In both cases, US imperialism was presented as wholly benign, concerned to disseminate the benefits of American liberty while preventing European colonial powers from expanding their influence across the globe.