Veselý was born on 24 April 1990 in Ostrava (Czechoslovakia then, Czech Republic now). His father, Jan, is a former basketball player, and his mother a former volleyball player. Veselý’s younger sister is a basketball player. Reportedly, he is fluent in Czech, Slovenian, Serbian and English.

Veselý began playing basketball with the youth clubs of Příbor and BK Snakes Ostrava in the Czech Republic. In 2007, he moved to Slovenia and signed with Geoplin Slovan.

In April 2008, he signed a multi-year deal with Serbian club Partizan. He won nine club trophies with the club, and also reached the 2009–10 Euroleague Final Four. In 2010, Veselý was presented the FIBA Europe Young Men’s Player of the Year Award.

Veselý was drafted in the first round by the Washington Wizards with the sixth overall pick in the 2011 NBA draft. After Flip Saunders was fired, Veselý saw more minutes on the court under new coach Randy Wittman, who also said that he wanted him to shoot the ball a bit more. On 9 April 2012, he recorded his first double-double, finishing the game with 11 points and 11 rebounds in a win over the Charlotte Bobcats. Veselý continued his success over the Bobcats on 23 April 2012 when he posted a career-high 16 points on 8-8 shooting. He averaged 4.7 points and 4.4 rebounds in his first NBA season. However, in his second season with the Wizards, he averaged just 2.5 points and 2.4 rebounds per game and while shooting just 31% from the free-throw line.

On 20 February 2014, Vesely was traded to the Denver Nuggets in a three-team trade. Until the end of the season, he appeared in 21 games, averaging 4.4 points, 3.7 rebounds and a career-high 1.3 steals per game.

On 5 August 2014, Veselý signed a two-year deal with the Turkish club Fenerbahçe.

Fenerbahçe advanced to the 2015 Euroleague Final Four, the first time in the team’s history that they played in a EuroLeague Final Four. On 15 May 2015, however, they lost in the EuroLeague semifinal game to Real Madrid, by a score of 87–96. Veselý contributed with 20 points and 6 rebounds in the semifinal game. Eventually, Fenerbahçe finished in 4th place in the EuroLeague, after losing in the third-place game to CSKA Moscow, by a score of 80–86. Over 29 EuroLeague games played in the 2014–15 season, he averaged a career-high 11.2 points and 5.4 rebounds per game, on 63.3% shooting from the field.

He was the EuroLeague’s 2015–16 season selection as the MVP for the month of January. On 21 March 2016, it was announced that he would be sidelined for up to six weeks, due to an Achilles tendon injury, which caused him to miss the entire EuroLeague playoffs against Real Madrid. He’d make his return on 13 May 2016, in the 2016 Final Four match-up against Laboral Kutxa. In his return, he’d record 14 points, in an 88–77 overtime victory against them, while also being announced as a part of the All-EuroLeague First Team, for his performances in the 2015–16 Euroleague season, before his Achilles tendon injury.

On 9 July 2016, Veselý signed a new three-year contract with Fenerbahçe, with NBA opt-out clauses every summer. In May 2018, he was named the All-EuroLeague First Team for the 2017–18 season, his second career nomination. In 2017–18 EuroLeague, Fenerbahçe made it to the 2018 EuroLeague Final Four, its fourth consecutive Final Four appearance. Eventually, they lost to Real Madrid with 80–85 in the final game. Over 34 EuroLeague games, he averaged career-highs of 12.5 points and 1.6 assists, while also having 5.1 rebounds per game.

On 18 February 2019, Veselý signed a three-year contract extension with Fenerbahçe. On 9 May 2019, Veselý was once again named to the All-EuroLeague First Team. Later that month, Veselý was named the EuroLeague MVP for 2018–19 EuroLeague season.