This is a powerful thing to realize – if you can talk to people the right way about what you are doing, you will become much more comfortable with street photography.
The goal of course is to not have everyone stop you. You can get close and shoot in pleasant ways that will generally keep people from asking you what you’re doing, but it still happens, and the key is to be obvious about it.
When someone stops me, I smile and flatter them. I tell them I’m a photographer doing a project on the area and I had to capture the scene with them as they looked great. I smile and act like I didn’t think I was doing anything wrong because I wasn’t.
This goes a long way and usually ends in a pleasant conversation. I sometimes offer to send them the photograph as well.
But in the situations where the person is still uncomfortable, offer to delete the photograph and apologize. Legally you don’t have to of course (in the U.S., U.K., and most of Canada at least), but it’s the nice thing to do.
On a similar note, I also want to highlight that the more comfortable you look and act while photographing, the more people will disregard you.
Over time, you will figure out a way of carrying yourself out there, which will make this type of photography much more comfortable.
People will notice you no matter what, so acting sneaky from afar will get their attention, while smiling, getting in the action, and looking like you’re having a great time photographing will put people’s guards down so that they leave you be.
And finally, the only way to fully get comfortable is to just to get out there consistently and push yourself a bit more each time. With enough of this, it’ll eventually become second nature.