*extra-*, as a Latin root, means “outside of, beyond”. So something that is extraordinary is outside of or beyond what ordinarily happens. A more literal modern English translation might be “out of the ordinary”.
The other usage, to mean “more of something”, comes from that same root. An *extra* cookie is another cookie *outside of or beyond* what you expected, paid for, etc.
*Super* has a similar origin. The original sense of the word is “above, higher than” as in a direction or a hierarchy. Gods (in particular the Christian god, in the case of the origin of this specific word) are both literally and figuratively *above* nature, and hence *super*-natural.
The modern usage of *super* to mean “very, extremely” comes from the sense of “above”, in the sense that “more” of something is (visually) a higher “pile”. The exact same process happened for its Greek equivalent, *hyper-* (which itself shares a root with Latin *super*).
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