Both Conor and I had similar observations — Conor recently, and me at about his age (now 5) or even older back in the day (Before the iPhone Era, forthwith known as BiE). Like I did to my parents at the time, he insisted that it was following us when I told him that it just looked like it was following us.
The simple explanation is that whether you're in a car, or more so by foot (as we were), the distance traveled is too little to affect how we perceive it. The moon is a big object that is very far away from us (relatively speaking). So if we move 100 feet or 1,000 feet here on Earth, it's not very much compared to the distance to our lunar satellite. So if you looked up and saw it as you left the house and then again when you crossed the street a mile down the block, it would appear to be the same size and in the same place — in other words, as close to you as when you started your walk, so it would appear to be "following," or rather "keeping pace" with you, rather than being left behind.
Other things, like clouds passing by, can enhance the illusion of the moon moving. As a kid, this seemed like a neat way to simulate a trip to the planetarium: I sat on the driveway, focused entirely on the sky, and if the winds were cooperating with a partly-cloudy sky, for a few seconds I could pretend I was flying.
Both a simple and a more detailed explanation (including the angle changes that affect perception), are available here on Rhett Allain's blog on Wired. I particularly like the video he made with Legos and his kid at the bottom of the post. Even if a younger kid may not understand the above explanation, his video makes it easier to understand the concept in general. Plus, Legos!