Why do Muslims consider their religion an extension of judaism and Christianity?

Answer 1

Islam traces its spiritual and ethnic lineage to the patriarch Abraham through Ismael, the son he had through his slave woman.

Ethnic: As the story in Genesis goes, there was tension between Ismael's mother and Sarah, Abraham's wife, throughout the boy's childhood. When their own son Isaac was born, the tension increased, and Ismael and his mother were turned out. Islam, Judaism, and Christianity all view Ismael as the patriarch of the Arabs, and therefore ethnically the spiritual father of Islam.

Spiritual: Mohammed, the founder of Islam, was himself Arabic. Islam developed first among the Arabic people, and the Q'ran contains many stories similar to those in the Torah (Old Testament). Since Ismael and his mother were monotheists, under the household of Abraham and Sarah (also believers in the one God YHVH), the monotheistic foundation of Islam is considered based in Abramic teachings (see Genesis).

Although not a Q'ran scholar, I did read the Q'ran in university, and I seem to recall it saying that Muslims should not bother other people "of the Book" (the Torah) -- meaning the other branches of YHVH-based monotheism -- Judaism and Christianity. But, as in all religions, believers do not adhere to the actual teachings and writings of their founders and the result is a very toxic planet.