It's a poetic way of referring to the past. It evokes a sense of nostalgia. It's a word most people understand, but don't normally use.
"Adown the lanes of memory bloom all the flowers of yesteryear"--Edgar Guest (a sentimental poet. In fact, a sentimental poet of yesteryear!)
"These were some of the old-fashioned ways the boys and girls of yesteryear passed a happy evening."
Googling for examples I found MANY references to variations on the phrase "Where are the snows of yesteryear?" ("Where are the clothes of yesteryear?" "Where are the wits of yesteryear?") I learn from Wikipedia that this is a famous line from a poem by François Villon, "Ballade des dames du temps jadis." The line is "Mais où sont les neiges d'antan?" and "This was translated into English by [Dante Gabrielle] Rossetti as 'Where are the snows of yesteryear?' for which he coined the new word yester-year to translate Villon's antan. The French word was used in its original sense of 'last yea', although both antan and the English yesteryear have now taken on a wider meaning of 'years gone by'".