Buddhism, Christianity, and Islam are three major universal religions (Judaism was the first monotheistic religion but is not a universal religion). The reason that these three faiths have shaped and continue to shape cultures around the world is that they spread as universal religions--that is, they promised that anyone who joined them could have access to salvation.
Unlike other faiths or pre-existing religions, Buddhism, Christianity, and Islam promise salvation to everyone, regardless of their color, pre-existing religion, gender, or social class (or other variables). The practitioners of the religions are not supposed to allow any intolerance; everyone can be accepted if they just accept the deity of that faith. By following the precepts of the founder of the religion (Buddha, Jesus, or Mohammed), anyone can get on a path towards salvation and turn away from acts that are considered evil. These religions are also not necessarily associated with one country or people, and they can be practiced in different languages and in different cultures. In other words, they transcend cultural barriers because they spread across cultures in the past.
Why do you think the essence of these religions has affected so many cultures around the world? It's because these faiths promise acceptance, and they put up no barriers to joining. They also promise salvation and a better life to the faithful, no matter what people's previous life or sins.