Instead of asking why is eye contact important, you should be asking why wouldn't it be important.
Looking into the eyes of a woman can be just as arousing as touching her.
How? Because it is a strong signal of flirtation, and also helps to build trust between people, which is very important.
This is also why many salespeople employ the use of eye contact in pitches.
Obviously you can't just run around all day through the city staring at every beautiful girl you see.
There is a fine line between flirtatious eye contact and creepy staring! It's best to start out slowly at first, with only short but fixated gazes.
As she becomes more trusting of you, you should notice her doing it back to you.
If all goes well and she is attracted to you, try holding the stare until she breaks it away from you.
If a gaze goes on for longer than 10 seconds, or generally just gets weird, it's time to act.
Either she wants you, or she's a nutter! Depending on how well you think things have been going, either stop talking and move in slowly for the kiss close, or do something silly to break the stare.
Poking your tongue out usually works like a charm! If you find it difficult to keep eye contact with a girl (or a guy for that matter), it may be worth your while to practice a bit.
In your daily tasks make eye contact with everyone you interact with.
Practice on not breaking first.
You will be surprised at how many women, even just at the supermarket, will find this a real turn on.
As previously stated, it shows confidence, and builds trust.
Don't be scared to do this with guys too, especially with salesmen.
You might get a better deal out of them? Try to mirror both men and women's reactions too.
If they smile, you smile.
If they wink, you wink.
If think try to kiss you, well..
Make a decision! Mirroring is one of the strongest trust building exercises you can do.