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Here is a "intuitive" answer, for a more in depth explanation of awk's mecanism see either @Isaac's or @Cuonglm's

Let's see,

 !a[$0]++ 

first

 a[$0] 

we look at the value of a[$0] (array a with whole input line ($0) as key).

 !a[$0] 

If a[$0] wasn't referenced before, it is assigned the empty string. It will evaluate to either "" (empty string) or 0 depending on context. ( ! is negation in test will evaluate to true)

• we print the input line $0 (default action).

• Also, we add one ( ++ ) to a[$0], so next time !a[$0] will evaluate to false.