Brodzinski attended Blaine High School in Blaine, Minnesota, where he was teammates with Florida Panthers center Nick Bjugstad. After high school, Brodzinski played two seasons with Fargo of the United States Hockey League before attending St. Cloud State University, where during his first season he led all NCAA freshmen with 22 goals. He finished his college career with 112 points (64 goals, 48 assists) in 120 games and was chosen by the Los Angeles Kings in the fifth round (No. 148) of the 2013 NHL Draft.
The 6-foot-1, 217-pound center was assigned to Ontario of the American Hockey League after signing a two-year contract with the Kings on April 1, 2015. He competed in the 2015 AHL All-Star Game before being called up by Los Angeles on March 23, 2015. He made his NHL debut against the New York Rangers two nights later, becoming the first player in Kings history to wear No. 76 and the first of the Kings' 2013 draft class to play for them. He scored his first NHL goal in a 4-0 win against the Florida Panthers on Nov. 18, 2017. He had six points (four goals, two assists) in 35 games with the Kings in 2017-18.