When you start a sentence with "Honestly..." it sounds like you're telling your listeners a secret that you wouldn't tell other people. For example:
Honestly, I don't think I ever want to have kids:
People usually use "Honestly..." to share things that some other people would disagree with. It can help you and your listeners to feel closer to each other.'
Another way to use "Honestly..." is to introduce an idea that you just thought of that has surprised you a little:
Honestly, I don't think I even need this jacket today.