Goosebumps are caused by a tiny muscle attached to each follicle, called the arrector pili. When you get cold or scared, these muscles involuntarily contract. This happens in other mammals, too, like a cat's puffy tail when it gets scared.
The problem is that wen you've shaved, the hairs are like miniature tree stumps instead of tapered blades of grass. When the arrector pili contracts, it pulls this little stump further out of the skin making it feel even more stubbly. I used to swim a lot, and sometimes these goosebumps actually hurt a lot!
So, that's what's going on-- more than you wanted to know, huh?
The only way to deal with this more effectively is to epilate. You'll still get the goosebumps, but the stumps won't pop out!