From the way he (= a chimpanzee) looked, it was easy to tell that he didn't want the nut, but he wanted to thank me for that.
Does the object of "tell" include "but he wanted to thank me for that" as well as "that he didn't want the nut"?
"tell" in this case is being used to mean "understand". The sentence means that, from the chimpanzee's look, it was easy to understand two things:
1. The chimpanzee didn't want the nut.
2. The chimpanzee wanted to thank me for that.
It is not clear from the sentence what "that" in #2 is referring to.
I don't know if that answered your question but I hope it helped.