Before the Empire’s rise, Chewbacca was a military leader for Kashyyyk, and somewhere along the way, he became adept with operating and repairing all sorts of technology, including starships. This ability is not surprising, as many Wookiees are technologically proficient. However, Chewbacca's strength with hyperspace technology hints at other adventures in his backstory that the Star Wars franchise has not covered yet.
Some clues to Chewbacca's past come in the Star Wars: The Clone Wars Season 3 finale, “Wookiee Hunt.” In the first episode in the arc, "Padawan Lost," Ahsoka is captured by Trandoshan hunters and forced to be prey alongside kidnapped Jedi younglings in the Trandoshan's hunting game. Then, in "Wookiee Hunt," Chewbacca also is kidnapped and is a pivotal part of the group's escape. Chewbacca is able to easily build a deep-space transmitter from scavenged parts, and his allies step in to save the group and bring them home.
In a featurette for the episode, Dave Filoni reveals that, while technically proficient, Wookiees did not have deep space-faring vehicles at the time of the Clone Wars, mainly because they did not have much need for them. Filoni does point out that they do have some ships; Yoda uses one of the Wookiee's pods to escape Kashyyyk after Order 66. Still, for most space travel, he explains that the Wookiees have contacts that they can rely on for transportation. In "Wookiee Hunt," Chewbacca's allies hire Sugi, a Zabrak bounty hunter, to take them to Wasskah to help Chewbacca, Ahsoka and the younglings escape.
This revelation about Wookiee culture makes Chewbacca's talent for piloting and repairing ships even more impressive. By the time of Solo: A Star Wars Story, he already is extremely adept at both skills. Han's surprise when Chewbacca reveals his abilities in the film also makes more sense when considering this information. Since the Wookiees were enslaved soon after the Empire's rise, it is unlikely that Chewbacca learned these skills after the Clone Wars. Instead, he probably learned them in the earlier decades of his life.
Fans have always known that Chewbacca has strong technical capabilities, but his knowledge of how to fly and repair the Millennium Falcon suggests that he has had a lot more experience with deep space vehicles than the average Wookiee. While some information from Chewbacca's past has been revealed in canon materials, such as his marriage to Mallatobuck and the existence of his son Lumpawaroo, "Wookie Lost" makes it clear there are more stories to be told about his life before he met Han Solo.