"他高兴得不知道说什么好"
He is so happy that he doesn't know what to say.
The grammar here is similar to the "so + adjective + that" form in English.
Here, 高兴得= is so happy that
After "that" or "得" comes the result. What's the result of him being so happy? The result is that he doesn't know what to say.
More examples:
他凶得没人敢和他做朋友。
He is so fierce that nobody dares to make friends with him.
他兴奋得睡不着觉。
He is so excited that he can't sleep.
考试难得大家都不及格。
The exam was so difficult that everyone failed it.
我饿得可以吃下十碗饭。
I'm so hungry that I can finish ten bowls of rice.