My baby loves to grab the pages and turn them. She learned how to do this when she was five months old. The key is propping up the page for her, as even an adult can have a hard time grabbing individual board book pages. So, when it's about time to turn the page, I prop it up a little with my right thumb or index finger, and she grabs the page and flops it over. When she was first learning this skill, she would just grab the page and flop it back and forth. I would guide her hand to show her how to turn it just once, and in the one direction. Gradually, she started turning the pages in a purposeful way.
At times she starts trying to turn the pages too fast, as if the fun of reading the book is just about turning the pages. When this happens, I will go back to a skill an expert nanny taught me: I tap on the board book page to draw her attention to the art. The nice thing about board book pages is that they make a good sound when you tap on them with your nail. So, that gets her to focus on the page long enough for me to finish reading it (or at least for me to say a few words, so the experience feels like reading).
Then she gets to turn the page and see what's on the next one! Yay baby!
Here is a video of my little page turner in action, when she first learned the skill (at almost exactly 5 months), a bit tentative:
And here she is at 9 months, my little speed demon page turner!