How did religion help shape the Israelites' culture as presented in the Torah?

Answer 1

The culture of the Israelites was the Torah.

The Torah and the Prophets played a vast role in the shaping of Israelite (Jewish) culture, because they are Judaism. They define it. Judaism entails living according to them. The Hebrew Bible reiterated the teachings of Abraham (Talmud, Yoma 28b) and codified their exact details; and at Mount Sinai, the Israelites accepted as obligatory what had until then been national custom. The Torah and Prophets have shaped our ways of thinking and provided a national legal code:
Practicing kindness and avoiding dishonesty became obligatory instead of merely proper.
Agriculture in the Holy Land included tithes to be given to the Levites and Kohens, thus providing for a scholarly class of people. One of the tithes was given to the poor, thus obviating the existence of starvation.
Immorality and incest were legislated against in detail. Instead of instinct or "crimes against nature," they were subsumed into religious law.

The roles of king, prophet, Kohen, Levite, officers and judges were all provided for in the Torah, thus defining the shape of the society and its institutions and providing certain balances.

The Israelite year was filled out with the Sabbaths and national festivals; and they were imbued with the function of worshiping God instead of being secular celebrations.

The judges were commanded to fear God, instead of relying on their skills of jurisprudence alone.

The laws of the Tabernacle (and later the Holy Temple), and commands to love God and fulfill all of the commandments, were written in the Torah together with (and mixed among) the seemingly mundane laws of testimony and witnesses (etc.), in order to convey the message that for us it is all part of religion. Secular life was a foreign concept. For example, a shopkeeper would be constantly aware of the religious laws of maintaining honest scales, giving a tithe to the poor (maaser kesafim), not overcharging, returning lost objects left behind, etc.; and he would set aside times for the daily prayers.
The above are just a few examples.

Hebrew (Israelite) society wasn't perfect. We are human; and as seen in the Prophets, there were times of backsliding and various pitfalls. But to the extent that the people kept their national laws and tradition, the society was exemplary and created a model which has influenced the Western world down to today.

See also:

How do our Jewish beliefs affect us?

Jewish life in Israel

More about the Hebrew Bible

Jewish history timeline

Answer 2

The laws or commandments tell them basically how to live in their culture.