Thank you for reaching out to Your Office Mom for advice. I'm happy you are looking at the internal job opportunities available to you! You are right in that there's nothing wrong with you looking for an new, internal opportunity. Companies encourage internal talent to apply for positions, so don't feel like you are doing something wrong. It's unfortunate you have to worry that your boss might take it personally. Who needs that, right?
In my opinion, all you need to worry about is how to break the news to your boss. Before she hears it through the grapevine, do it as soon as possible. I suggest you give her a heads up and in the process, try to reduce her angst, to (as you say) preserve the relationship.
In a private setting, just say in your own words, "I wanted to let you know that I've applied for the ______ position in _______. Since I've been here for ___ months/years, I felt like it was time to explore my career opportunities. I saw the posting and it looked good/interesting/perfect because of _______ and ________.
This approach is straightforward, and it explains your rationale (not that it's needed) to minimize any defensiveness on her part.
Good luck to you in your job search!