I have seen another transgendered person several times in shops near where I live, and haven't yet plucked up the courage to talk to her
You're absolutely 100% sure she's trans right? You've seen her ID and it's a male name and gender designation with her picture on it? You've looked under her skirt and discovered she pre-op or non-op?

Assuming you're right and she is/was TG, what are you planning on talking about? If you see her regularly is she living full time? So you want to do what, help her dredge up a painful past? Perhaps she's early in her RLE and her confidence is still somewhat fragile now you come along and, not meaning to of course, shatter what confidence she has built up?

How would you approach any other stranger you meet? She's no different and should be treated as you would any other woman. What would you say and why?

I know this is a bad example but it's the best I can come up with at the moment. If you had siezures and knew someone else did too but you didn't know them personally would you be drawn to talk to them and ask them about their seizures?

She is a woman and a person just like anyone else and deserves to be treated as such. If she's standing on a soap box proclaiming to the world she's trans that's one thing. If she's just trying to live her life like any other woman, like any other person, that's a completely different thing.