IanOji's writeup is not entirely accurate.
The concept of mono no aware has been translated by the scholars of japanese literature as "exclamation of sympathy or distress" (Arthur Waley), "the pity of things" (Ivan Morris), and "the sad transience of things" (Stephen Shaw), among others. The concept of the transience of the world is a fundamental principle of Buddhism, which was gaining traction in the Heian period (from which the concept of mono no aware originates)
So where does the concept of "sadness" or "pity" come from? Mono no aware, in japanese, is written
物の哀れ. Using the kanji "哀", we see that the reading for "aware" (roughly pronounced ah-wa-rey) is not the verb 'au' but an adjectival noun meaning "helplessness, pathos, pity, sorrow, compassion".
While the literal meaning of the phrase is not difficult to understand, the phrase's "untranslatable" nature is a result of the cultural nuances it evokes.