It depends how you intend the question.
If the question is asking why Jews debated religion with each
other, it is simply because they did not always agree with each
other. Judaism encourages religious dissent and discussion of
different religious views.
If the question is asking why Jews debated religion with
non-Jews, it was usually because they were compelled to by the
Christian or Muslim monarch. It was meant to show that Christianity
or Islam was the proper replacement for Judaism as the Divine
Religion. It was a lose-lose for the Jews who engaged in such
debates since the Jew could either hand the debate to the non-Jew
and "prove" the point that Judaism was inferior or win the debate
and risk a pogrom against the Jewish population in retribution for
"blaspheming" the dominant religion. The latter was the result of
the famous debate called "the Disputation" in the Kingdom of Aragon
between Pablo Cristiani (Catholic) and Nahmanides (Jew).
