Here's why Harry doesn't use magic:
* He's underage: Harry is still 11 years old in this book, and the law prevents him from performing magic outside of Hogwarts.
* The statute of secrecy: This law exists to keep the magical world hidden from Muggles. Using magic in front of Muggles would be a serious breach and could endanger the entire wizarding community.
* Dumbledore's warning: Dumbledore specifically warns Harry against using magic outside of school.
The exception: The only time Harry uses magic during the summer is when he is fighting off Dobby the house-elf. This is a desperate situation, and Harry is using magic to defend himself.
Important Note: Although he uses magic, he doesn't realize he's doing it. Dobby is preventing him from casting spells, but his magic is still being used subconsciously to try and fight back.
This situation highlights the dangers of trying to suppress magical ability, and how it can lead to unintended consequences.