The main reason Lady Jane was executed was because she was a potential claimant to the English throne, and therefore a threat to Queen Mary as long as she lived. There are a couple layers to this:

Firstly, Henry VIII dictated in his will that if Edward VI should die without issue, then Queen Mary should inherit the throne (Ives, "Henry VIII's Will-A Forensic Conundrum", The Historical Journal, December 1992, pp. 779-804). Edward was a boy just shy of sixteen when he died without an heir in July of 1553. Instead of the crown passing directly on to Mary as was the will of Henry VIII, Edward tried to control the passing of the crown in a similar manner to his father. Edward tried to ensure that the crown would pass to Lady Jane, his cousin (She was the daughter of Henry VIII's younger sister), who was a committed protestant (Ives, "Tudor Dynastic Problems Revisited", Historical Research, May 2008, p. 269). Lady Jane was proclaimed Queen in London, but her "rule" lasted less than two weeks, as the majority of the nobility and commons supported the much more popular Mary as the heir for the English crown. Mary also was able to assemble enough support from the elite in East Anglia to threaten London with the use of force, and this pushed the city and Council to proclaim her queen (Ives, Lady Jane Grey: A Tudor Mystery, part III; Loach, Edward VI, pp. 170-179).

Jane was arrested and sent to the Tower of London as Mary was proclaimed Queen across England. The Imperial ambassador wrote the day after the proclamation (July 20) that "On that occasion the people showed unspeakable joy, both in public, in private and in the streets, shooting off salvoes of artillery and indulging in the accustomed sports and games. On the other hand, when [Lady Jane] was proclaimed, there was no pretence of rejoicing." (Calendar of State Papers, Spanish, Volume 11 (1553) p. 104). Initially, Queen Mary intended to free Lady Jane from her imprisonment in the tower, after she felt assured that Jane would not be a danger. Jane was put on trial for treason in November of 1553, though the Imperial ambassador was assured that Jane would be pardoned (ODNB Online, "Lady Jane Grey" by A. Plowden). In late January of 1554, while Jane was still held in the Tower, Sir Thomas Wyatt attempted to lead a protestant inspired rebellion against their new Catholic queen, who also happened to be proceeding along in marriage talks with Philip of Spain (Thorp, “Religion and the Wyatt Rebellion of 1554” Church History, December 1978 pp. 363-380; Fletcher & MacCulloch, Tudor Rebellions 5th ed. pp. 98-102). Jane’s father had been pardoned for his role in the events of July 1553, but he participated in Wyatt’s rebellion. Jane was seen as the possible figure head – the protestant answer to Mary. Jane may not have been responsible for the revolt, but her cooperation in the succession conflict in 1553, her possible claim to the throne, and the threat of popular discontent against Mary’s government sealed Jane’s fate. She had become too much of a danger to Mary and to the stability of the nation.