Lee was raised in Edina, Minnesota, a suburb of Minneapolis, to Thomas (Tom) and Lisa Lee. He excelled in football, ice hockey and baseball from a young age. In 2004, he transferred to St. Thomas Academy, a private, military high school in Mendota Heights. He made the varsity ice hockey team as an eighth grader and played alongside Jordan Schroeder. He was also part of St. Thomas Academy's Class A high school hockey tournament, winning the championship in 2006 and playing again in 2007.
In the fall of 2007, Lee transferred back to Edina High School. Although private-to-public school transfers were rare, the daily 30-mile round trip commute to the St. Thomas campus and a desire to play with teammates he grew up with factored into his decision. In his junior year, he made an immediate impact in both football and hockey. In the fall, he won the starting quarterback position and passed for 2,049 yards and 14 touchdowns for the Edina Hornets, who finished with a 10–1 record after losing to the eventual state champion Eden Prairie in the Section 6AAAAA championship game. This was the best finish for the Edina Hornet football team in the past decade. In the winter, he transitioned to hockey, where he finished third in scoring for his team with 54 points (32 goals, 22 assists). He was selected to the Pioneer Press All-State First Team. and received an honorable mention to the AP All-State team. He helped lead the hockey team to a 26-1 regular season record and a second-place finish in the Class AA 2008 state hockey tournament. In the spring, he played varsity baseball as both a pitcher and third basemen.
As a senior, he had a breakout year in football. As quarterback, he passed for 2,007 yards and five touchdowns and ran for 1,105 yards and 32 touchdowns, averaging 308.7 yards of offense per game. He was selected as the 2008 Minnesota Gatorade Football Player of the Year, the Star Tribune All-Metro Player of the Year, and was a finalist for Minnesota's "Mr. Football", losing to now professional player, Varmah Sonie. He holds the state record for most all-purpose yards in a game, gaining 581 yards (477 passing and 104 rushing) in 42–56 loss to Hopkins High School. Despite the individual on-field success, the Hornets went 6-4 that year and did not make the playoffs.
Lee's success in football carried over to the ice, where he had another successful season for the Hornets ice hockey team. He finished as one of the top scorers in the league during his senior year, scoring 25 goals and 59 assists in 31 games. He led the Hornets to a 24-3 regular season record and 2009 high school hockey tournament appearance. Despite being the second seed in the tournament bracket, the Hornets lost in the opening round to the eighth seed Moorhead High School. The Hornets would go on to win the consolation final, finishing fifth-place overall in the Class AA tournament. He was a 2009 finalist for the annual "Mr. Hockey" Award, losing to now professional hockey player and future Islanders teammate, Nick Leddy. Playing in the pre-season Upper Midwest Elite League, Lee scored 12 goals with 17 assists in 18 games for Team Southwest. Many scouts regarded Anders as one of the best athletes in the state of Minnesota. Some even believed he was the best in the nation. While many hoped he would sign to play football and hockey at the University of Minnesota, Lee ultimately committed play hockey at the University of Notre Dame.
In the 2009 National Hockey League (NHL) draft, Lee was selected 152nd overall by the New York Islanders. He spent the upcoming season playing for the Green Bay Gamblers in the United States Hockey League (USHL) and was their leading scorer. He scored 35 goals with 31 assists and had a team-best +38 rating with 54 penalty-in minutes in 59 games and was chosen for the 2010 All-Star Game. He was named Playoff MVP after scoring 10 goals with 12 assists and finishing +10 in 12 playoff contests. He was also named "Rookie of the Year" for the season. The following season, Lee was the second-leading scorer for the Notre Dame Fighting Irish of the Central Collegiate Hockey Association (CCHA) as a freshman as the team were one of the turnaround stories in college hockey, finishing second in the league after finishing ninth the previous season. He scored a team-leading 24 goals with 20 assists and his 44 points were topped only by fellow freshman T. J. Tynan, who had 54 points. After falling in the CCHA playoff semifinals and consolation games Notre Dame rebounded to capture the NCAA Northeast Region and advanced to the Frozen Four, losing to eventual national champion Minnesota-Duluth in a semifinal game. In the 2011–12 season, Lee was the second-leading scorer for Notre Dame as a sophomore. He scored 17 goals with 17 assists and was minus-one with 24 penalty minutes. The Fighting Irish finished eighth in the league, reaching the tournament quarterfinals against Michigan after defeating Ohio State in an opening round series.
Lee received his first call up to the Islanders during the 2012–13 season. On April 2, 2013, he scored his first NHL goal in the opening period of his NHL debut, against Ondřej Pavelec of the Winnipeg Jets. He was sent to the team's minor league affiliate, the Bridgeport Sound Tigers of the American Hockey League (AHL) for the following season, but was called up to the Islanders again shortly after the midway point in the season. He began the 2014–15 season with the Sound Tigers but was quickly called back up for a third time to play with the Islanders in a more long term situation.
Lee had a career year in the 2016-17 season, playing mostly on a line with John Tavares and Josh Bailey.