Why was Hamas established?

Answer 1

Hamas was created to destroy Israel. They tried to destroy Israel since 2006 but they did not come close. They bombed Israel over 50,000 times and only 131 people died.

Hamas was founded by Sheikh Ahmed Yassin, a Palestinian spiritual leader who became an activist in the local branch of the Muslim Brotherhood after dedicating his early life to Islamic scholarship in Cairo. Beginning in the late 1960s, Yassin preached and performed charitable work in the West Bank and Gaza Strip, both of which were seized by Israeli forces following the 1967 Six-Day War. In 1973, he established al-Mujamma' al-Islami (the Islamic Center) to coordinate the Muslim Brotherhood's political activities in Gaza. Eventually as Fatah and other secular-aligned Palestinian militant groups were less and less successful, the Muslim Brotherhood stepped into the breach as a theocratic-aligned Palestinian militant group. The rest, as they say, is history.

Answer 2

It depends on how the question is read.

Morality
If it is asking from a moral perspective, e.g. "Why is it a positive thing that Hamas exists?", I really have no answer. All of the positive things that Hamas does, such as administer schools, hospitals, and community centers for Palestinians in the Gaza Strip are more than offset by their negative actions towards Palestinians, such as allocating all incoming funds to the war against Israel instead of regional economic revitalization, telling civilians to go into buildings that Israel has said it would bomb, firing over the backs of Palestinian children to kill Israeli soldiers, and numerous other atrocities. (This is, of course, ignoring Hamas' destabilizing and violent effect over the border in both Israel and the Egyptian Sinai Peninsula, which are unequivocally bad.)

History
If it is asking from a historical perspective, e.g. "Why is it natural or expected that Hamas exists?", this is really straightforward. There were two simultaneous events going on the Palestinian community in the late 1970s and early 1980s.

1) Rise of Islamism: The first was the general rise of Islamism across the Islamic World. Islamism was the belief that Islamic Law should be the national law of governments across the Arab World, as opposed to secularism, which believed that religion should stay out of government and was prevalent in the Arab World in the 1960s. The Palestinian Muslim Brotherhood, which was an Islamist organization was able to take advantage of this rise in popularity of the Islamist brand, swell its membership, and create a militant off-shoot: Hamas.

2) Failure of Secularist Palestinian Groups: Among the Palestinian resistance, nearly all of the major groups were secularist, like Fatah, the DPLP and the PFLP. All of them had been unsuccessful in causing any major dent in Israeli policy. Until the 1980s, every Palestinian group was consistently losing territory, consistently losing membership, and consistently losing the public relations war. As a result, many Palestinians believed that a different group, like Hamas that was an Islamist group, might be successful where the other Palestinian resistance groups were not.