Why do men and women sit separately at traditional Jewish services?
All Jewish practices have their simple reasons as well as deeper, more spiritual explanations.
One obvious benefit of separate seating in a synagogue is that it helps ensure that the main focus is on the prayers and not on the opposite gender. There is no question that we don't act the same in a mixed crowd as we do in a same-gender one. There is nothing wrong with that. It is good and healthy that we are attracted to each other, but during prayers we shouldn't be trying to impress anyone other than G‑d.
In addition to that, a synagogue should be a welcoming and inclusive place. No one should feel left out. Many single people feel extremely uncomfortable at a function or event at which everyone seems to be with a partner except them. No one should ever feel this way at a synagogue. When men and women sit separately, there is no discrimination between singles and couples. (There will always be a chance for singles to mingle afterwards at the Kiddush!)
But it goes deeper than that. Women and men are very different beings. Not only are we physically different; our thought processes, emotional states and psychology are all different. This is because our souls are different - they come from complementary but opposite sources. The prayer experience is supposed to be an opportunity to be with your true self, to communicate with your soul. Men and women need space from each other to help them become intuned to their higher selves.
Ironically, it is by sitting separately in prayer that we are able to truly come together in the other areas of our lives; because it is only when both male and female spiritual energies are allowed to flourish that we are complete as individuals, families and a community.