It is derived from the actual modern Hebrew word for "a person from Israel", ישראלי, which is transliterated as "Israeli". The -i suffix is a common one in Biblical and Modern Hebrew to describe a person belonging to the same group. Other examples include פְּלִשְׁתִּי (pilishti - Philistine), and אֱמֹרִי (emori - Amorite). It is basically the same thing as the English -ite suffix. In the Hebrew Bible, the Hebrew people were called בְנֵי יִשְׂרָאֵל, or "Children of Israel". "Israelite" is a modern translation of that which is Anglicized (coming from the French and Latin). Typically, "Israelite" denotes the ancient group, specifically those from the northern kingdom of Israel, while "Israeli" denotes a citizen of the modern state of Israel, which has existed since 1945 1948 AD.

Edit: error correction