Readers hear the narrator speak in her own voice and, if she has an interesting voice, they feel attached to her right away.
It's like sitting across the table from her as she tells the story. You hear her side of the story, and you are prone to believe her. If she's cute and in trouble, you're likely to fall in love with her. You'll want to see her win against the villain.
For the author the advantage of writing with a first person narrator is that you get to know your character really well. You don't have to wonder if she's afraid. She come right out and tell you, "I'm scared. Get me out of here." Just as the readers feel like they know her better than they know a third person narrator, you also will feel like you know her better. There is just something about hearing a person's voice that makes us feel like we know her.
A fun thing about first person narrators is that they aren't always honest. They are telling their story as they see it and if you want them to feel like real people you have to let them have their own prejudices and blind spots.
Using the first person narrator is hard for several reasons. First you have to make sure you have a character that really does have an engaging voice. You're going to live with this character for a long time. Can you stand her? Will her sarcasm grow old and turn into obnoxious complaining after a few chapters? Will she run out of witty things to say before the book is done? If you choose a first person narrator make sure she's the kind of gal you want to listen to through the whole book.
Because she has to be in every scene. Which brings up the second limitation for an author using a first person narrator. You can't show us anything that your POV character doesn't see or hear or read about. What she knows, the reader knows. What she feels, the reader feels.
This is a lot harder than it appears. We are used to watching TV and movies and the camera is never in the character's head. It's always outside the character, roaming around. It can show you, in a wide-angle shot, things the character cannot seeŚlike the killer hiding behind the door. Or the guy sticking out his tongue behind her back. In a first person narrated book, that stuff is not there. You can't show it.
So weigh the pros and cons. Is your character quirky with a New York attitude? You may need to let her speak for herself. Is she quiet and thoughtful? She probably needs a narrator to tell us what she's thinking and feeling. And what about your plot? Are there things we need to know that the main character can't find out about? Then you can't tell the story in her POV exclusively.
Thinking through these things ahead of time will save you trouble down the road.