It is an invented word to illustrate the idea of words that describe themselves. It is constructed along regular lines (or almost). Awkwardness itself, a word that is generally recognised and included in dictionaries, is constructed in such a regular way. The adjective is awkward, which I shan't bother to define - it being in every reasonable dictionary. Awkwardness is constructed in the same way as fullness, emptiness, sadness and so on, and is a noun representing the characteristic of being awkward. If something is awkward, it has awkwardness. So, if one is in an awkward situation, one might comment on the awkwardness of the situation.
Now, we can construct new adjectives from some nouns, nouns that represent something that things might be "full" of - so a situation full of mirth is mirthful, and something that is prone to changes is changeful. The same suffix is also used to indicate quantities, the quantity of something that will fit into something. So an armful is the quantity of something you can carry with an arm (or the quantity of something that is in one arm), and a bowlful is the quantity of something that will fit in a bowl. However, it is the "being full of something" that is being evoked by this word.
Thus, awkwardnessfull (which ought to be awkwardnessful to fit the pattern, but that's a small niggle1) is an adjective describing something that is full of awkwardness. To a fluent speaker of English, the intention is obvious, and it's not unusual for writers (or scientists) to make up words in this way, with suffixes, prefixes and so on.
In the end, *awkwardnessfull essentially means "very awkward", but you won't find it in many dictionaries. You have to work out what it means by recognising the suffix.
The author is trying to be clever, make a point by doing something strange. That's the whole point of it.
1: Actually, -full was an alternate version of the suffix in Middle English, so maybe it's being rustic rather than not-quite-right. Or maybe not.