How does reduplication change the meaning of adjectives? What is the difference in meaning of:
它的眼睛大大的。
它的眼睛很大。
and
他有大大的眼睛。
他有大眼睛。
And more generally, when might someone use reduplication for adjectives?
Reduplication of adjective put emphasis on it

Same as English: "big" in "He has big big eyes" is more emphasized than the simple "big" in "He has big eyes" 

他有大大的眼睛 = he has very big eyes

它的眼睛大大的 = his eyes are very big

Notice, not all adjective can be reduplicated, for example we do not reduplicate 巨 because 巨 is already a higher superlative of 大

Also, we don't reduplicated adjectives that are made up of two same characters, For example, 茫茫 is a compound word, not a reduplication of 茫; 忽忽 is a compound word, not a reduplication of 忽