Khrushchev gave his speech to a closed session of the 20th Congress of the Communist Party of the Soviet Union on 25 February 1956. The contents of the speech was subsequently disseminated to a select group by being read to groups of party activists and “closed” local party meetings.
Even though knowledge of the speech was limited to a select group, that was enough to cause a sense of shock and disillusionment throughout the Soviet Union and the Soviet bloc. It damaged Stalin’s reputation and, more importantly, it damaged the perception of the political system as a whole.
In June 1956, Mátyás Rákosi, was forced to "resign" as General Secretary of the Party under pressure from the Soviet Politburo. Rákosi had previously described himself as "Stalin's best Hungarian disciple" and "Stalin's best pupil". With the denunciation of Stalin, Rákosi's days were clearly numbered. He was replaced by his former second-in-command, Ernő Gerő.
After Rákosi's resignation, students, writers, and journalists became more active (and more critical!) in politics. Students and journalists started a series of intellectual forums discussing and debating the problems that Hungary then faced. The forums became incredibly popular with thousands of participants. The debates generated protests, and those protests grew. They came to a head in October 1956.
The Hungarian Revolution began on 23 October 1956. It was driven by the students who had participated in the forums.