There are a number of reasons you can cite. You might want to check in your textbook to see which of these it says are the three.
First, there was the idea that Americans should support other people who were trying to become independent. The US was created by rebelling against a European colonial power and so we should support others in the same situation.
Second, there was the idea that the Spanish were brutal. Americans felt that Cubans were being terribly abused by Spaniards like Valeriano Weyler. Therefore, they felt, the US should come to the aid of the Cubans.
Third, "yellow journalism" helped to create interest in Cuban independence. Journalists like William Randolph Hearst pushed for the US to get involved by playing up supposed atrocities like those committed by Wyler.
Fourth, Cuba was part of what America saw as its own sphere of influence. Americans felt that an independent Cuba would be influenced by the US. This would remove the influence of a potentially hostile foreign power from an area so close to the US.
One key to understanding the position of many Americans with regard to the Cuban struggle for independence throughout the latter third of the 19th century is to harken back to the so-called "Monroe Doctrine" of 1823. Then-President James Monroe, in his annual State of the Union address to Congress and the nation, declared that the United States would resist any efforts by European colonial powers to extend their influence into the Western Hemisphere. All students of the American Revolution understand the role played on both sides of that struggle of European powers, specifically, the British, against whom the struggle was waged, and the French, who supported the revolution for their own foreign policy purposes and who continued to maintain an active interest in North America. The United States of America was born out of a rejection of imperialism, although, in the case of the Caribbean and Central America, that posture would succumb to greater geopolitical and commercial interests. Most Americans, consequently, were viscerally opposed to colonialism, and the continued Spanish occupation of Cuba stood as a constant reminder that the last vestiges of European colonialism in the Western Hemisphere had yet to be removed. As Cuban independence movements agitated, and fought for independence from Spain, therefore, they found enthusiastic support among those Americans--support galvanized in no small part due to the alarmist editorials regularly featured in major American newspapers warning of the dangers of the Spanish fleet anchored near U.S. shores.
American opposition to European colonialism in the Western Hemisphere, then, is one of the major reasons for American support for Cuban independence. A less altruistic reason for that support involved U.S. commercial interests in Cuba. Being only 90 miles from U.S. territory (i.e., the state of Florida), and a major agricultural supplier of tobacco and cane sugar, the Cuban and American economies were closely linked, which would prove a major vulnerability for the former. While the growth of the tobacco industry in the American South had supplanted Cuba as an important source of that particular commodity, the market for Cuban sugar remained very robust. It was in the U.S. interest, therefore, that the Spanish Empire be permanently removed from Cuba.
A third reason for American support for Cuban independence was also not particularly altruistic. As the United States grew economically and militarily, it sought to expand its reach so as to be able to protect its overseas interests and the flow of trade into and out of key ports along the American South. Many Americans are aware of the existence on Cuba of a U.S. Naval installation at Guantanamo Bay. The origins of that base lie in the outcome of the Spanish-American War. With the Spanish defeated, the victorious United States imposed upon Cuba an arrangement whereby the U.S. would lease an expanse of land for use as a maritime installation. The United States--keeping in mind the situation that existed in the late-19th-early-20th centuries--was concerned about its ability to protect the vital sea lanes that run between Cuba and the island of Hispaniola (Haiti and Dominican Republic), known as the Windward Passage. The U.S. Navy base at Guantanamo Bay, therefore, was constructed to enable the United States to protect that sea lane while enabling the United States to better project maritime power abroad.