Why are the ten condmanmenrs influencing many people around the world?

Answer 1

Because they are from God (Exodus ch.19-20), and because they are instinctively compelling.

The Ten Commandments are a pillar of Western civilization. They have improved the conduct of billions of people; but because of His covenant with them (Exodus ch.24), God specifically honored the Israelites with being the nation which would be the ones to receive and treasure the Ten Commandments (Exodus ch.19-20) until they had been absorbed by humanity in general. These Commandments are so important that God revealed His presence to the entire Israelite nation when He spoke these Commands. And He inscribed them in stone (Deuteronomy ch.9); unlike the rest of the Torah, which He had Moses write (Deuteronomy 31:24).

What would Western society look like without these Commandments?

1) "I am the Lord your God..." has shaped the Western beliefs about God. Pantheism and polytheism, which were excuses for immorality, are out.
2) The 7-day week, including a day of rest for everyone, is thanks to the Fourth Commandment.
3) Without the Fifth Commandment, society might still permit abandoning our aged parents to die.

4) Without the Sixth Commandment, we might still be killing our own children, as was done in Classical European nations until Judaism and its daughter-religions put a stop to it. The above are just a few examples.

And how do Jews live by the Commandments in our daily lives?
1) "I am the Lord your God" is our reminder to be aware of God constantly. The very first paragraph in the Code of Jewish Law (Shulchan Arukh) reiterates this command.
2) "You shall not have other gods" means not to make anything other than God too important. Example: not to "worship the almighty dollar" or anything other than God.
3) "You shall not use God's name needlessly" teaches us to deal with God's names with the utmost reverence. Other than in prayer and reading the Tanakh, we don't pronounce any of God's names at all.
4) "Remember the Sabbath day": to learn the halakhot (laws) of Shabbat (Sabbath) and to prepare for the holy day properly (nice clothes, food ready, etc.). The Sabbath is a testimony to the Exodus and to the Creation of the world (Exodus 20:10; Deuteronomy 5:14).
5) "Honor your father and mother": to do our best to honor and assist them; and to avoid being cheeky to them. No "generation gap" should be allowed to spoil family ties.
6) Do not kill: also hints at avoiding character assassination or endangering someone. Slander is forbidden in Judaism even when it's not untrue.
7) Do not commit adultery: flirting is playing with fire.
8) Do not steal: even a paperclip should not be filched.
9) Do not bear false witness.
10) Do not covet: to avoid jealousy and gazing longingly at what is not ours. To realize that the other fellow's possessions have been allotted to him by God.

See also:

How has the Hebrew Bible influenced Western society?

What is the history of the Hebrew Bible?