The novelization of Star Wars: The Last Jedi by Jason Fry explains why Luke Skywalker changed robes soon after meeting Rey on Ahch-To, making his arc in the film even more tragic. The Force Awakens famously ended with Rey arriving on Temple Island, where she finds the missing Jedi Master and presents Luke with his family's blue lightsaber. Picking up immediately where J.J. Abrams' film left off, Rian Johnson's sequel showed the aftermath of this fateful meeting: a befuddled Luke accepts the lightsaber, tosses it over his shoulder and storms off. His destination is his hut, where he quickly changed from the white and beige Jedi robes to the darker, Earth-toned outfit he wears for most of the movie.
Why Luke changed clothes is a minor mystery in The Last Jedi movie, but there is a reason for it, one thats tied to the ancient tree which housed the sacred texts of the Jedi. Luke's white robes are ceremonial Jedi apparel that are part of the ritual of burning down the tree. It turns out the last Jedi had long contemplated the act of lighting that tree ablaze, and with it, the remnants of the ancient Jedi Order, which he felt had been the cause of so much tragedy throughout the galaxy; something he was grappling with when Rey turned up.
The book makes clear Luke had actually made a few failed attempts to burn the Jedi library, but something prevented him each time. In the novel, Skywalker reflected that part of the reason was that he had spent the decades since the Battle of Endor traveling the galaxy with R2-D2 collecting Jedi lore and artifacts; this voyage of discovery meant the temple was his ultimate life's work.
When Rey found Luke on the cliff, she caught him in the midst of yet again being unable to destroy that tree, explaining why he was in the ceremonial robes and immediately changed afterward. The young Jedi-to-be then utterly surprised him by appearing with his family's lost lightsaber. Since Luke couldn't bring himself to burn the temple and he now had an unexpected visitor, the point of wearing the white robes was gone, so he retreated to his hut and changed clothes.
This is why Luke later dons the pale getup after Rey left the planet, disappointed in him; he was once again going to try and destroy the texts. Yet Luke faltered one final time and still couldn't light the flame. Instead, Yoda appeared as a Force Ghost to offer his former apprentice sage words of advice before using the Force to call down a bolt of lightning to set the tree ablaze at last. Of course, Yoda knew Rey had the ancient Jedi texts in her possession aboard the Millennium Falcon, so this wasn't quite the final stamp Luke believed it to be.
For Luke Skywalker, clothes do make the man - to a degree.