Doctor Who relies on the gimmick of a regenerating title role, asking different actors to take on the title of Doctor over its incredibly long run. Taking over a role from David Tennant is a big ask, but Matt Smith's intense and quirky Doctor showed he was more than up to the challenge. Like the Doctor before him, Smith bowed out after three seasons with the show, ready to tackle new challenges in his career but already regretting leaving the Doctor behind.
Matt Smith's Eleventh Doctor was a more inquisitive version of the role than many had seen. Smith was notably the youngest actor to play the part, and he leaned into his youth by emphasizing the Doctor's curiosity and lack of cynicism. But more than just his youth set Smith's Doctor apart -- never has a bow tie looked cooler than gracing the neck of his Doctor.
Smith joined the cast of Doctor Who at the same time as new showrunner Steven Moffat. The team brought a new interpretation to both the character and the series. Smith took on the mantel of the Doctor from the previous actor David Tennant in "The End of Time," establishing his new quirks and finding his companion in the following episode "The Eleventh Hour." Moffat, previously a Doctor Who writer, assisted in the production of the final scene of "The End of Time," and took over as showrunner for all of his following episodes. The clean break before the Smith/Moffat era of Doctor Who allowed the show to establish a new aesthetic with the Eleventh Doctor.
Smith stayed on as the Doctor for three seasons, from 2010 through the end of 2013. He announced his plans to leave Doctor Who mid 2013, giving no specific reasons for the end of his tenure in the role beyond just knowing it was time. An article in the Telegraph describing Smith's departure seems to indicate that he was on good terms with the showrunner, as both Smith and Moffat deliver effusive compliments to each other. While previous actors had left due to disagreements among the cast and crew, it seems as if Smith's choice was based on no such incidents.
The Doctor is a demanding role, however, which some have speculated contributed to Smith's departure after three seasons. The actor is on record with the Hollywood Reporter describing how intense the process of filming Doctor Who can be, and how after three years it was "the right time" for his departure. Other actors, including the Tenth Doctor's David Tennant, similarly described the intensity of the role, perhaps leading to the shorter seasons for the current Doctor.
Despite the demands of the job, Matt Smith enjoyed his time as the Doctor, even telling Digital Spy that he regrets not staying on longer to work more with actor Jenna Coleman, who played companion Clara Oswald in Smith's final season. His choice to leave, however, paved the way for his continued career, including his Primetime Emmy Award-nominated role of Prince Philip on Netlix's The Crown. Smith helped shape Moffat's interpretation of Doctor Who, but the show was only a stepping stone for the actor.