What is the story behind Robert Langdon's Mickey Mouse watch?

The vintage, collectors edition Mickey Mouse watch was a gift from Langdon's parents on his 10th Birthday. As per the book, Langdon has not owned any other watch after that.

Quoting directly from the book:

Although its juvenile dial often drew odd looks, Langdon had never owned any other watch; Disney animations had been his first introduction to the magic of form and color, and Mickey now served as Langdon's daily reminder to stay young at heart

Growing up after his introduction to the magic of form and color, when he started taking interest into symbols and like, he would have figured more about Walt Disney, who, he says that, was doing his best to pass on the Grail story to future generations, during a conversation with Sophie.

Again quoting from the book, about the further details he gave to Sophie about Walt Disney

Throughout his entire life, Disney had been hailed as "the Modern-Day Leonardo da Vinci." Both men were generations ahead of their times, uniquely gifted artists, members of secret societies, and, most notably, avid pranksters. Like Leonardo, Walt Disney loved infusing hidden messages and symbolism in his art. For the trained symbologist, watching an early Disney movie was like being barraged by an avalanche of allusion and metaphor.

This would have ensured that Walt Disney was and will always be a hero for Robert, and especially this watch would have been holding a very special place in his heart, which he said that introduced him to the world of magic, in the first place.

Apart from the already existing answers mentioning the books, the watch is also adressed a little more in Inferno, the third film in the series.

During Robert's kidnapping at the beginning of the story he loses his watch on the street (but gets it back at the end). When he notices that he's missing it in Sienna's apartment, he mentions that it's quite important to him and was a gift from his parents.