Born in Ballerup, Breschel got his breakthrough with small Danish Team PH, finishing 6th at the U/23 Cycling World Championship in Verona in 2004 where he helped fellow Dane Mads Christensen finish 3rd. He also won the bronze medal at the Danish National Road Racing Championship during the summer of 2004.
He turned professional for the 2005 season in Denmark based Team CSC, where he signed a two-year contract. At the press conference, regarding his choice to join Team CSC in October 2004, he stated that he simply wished to adjust to the rigors of professional cycling, saying "I hope to get in the team, but in the beginning I just want to learn the game and to learn the races. Somewhere I know that I'm in for a beating." Under tutelage of seasoned veteran Lars Michaelsen, Breschel would start the season in the Tour of Qatar, where the two riders finished side by side, Breschel conceding the final victory to Michaelsen. They would ride a number of classics and smaller races together, and Breschel finished in a number of secondary placings, just missing the victory podiums.
For the start of the 2006 season, he once again showed himself in Tour of Qatar, finishing as the best young rider of the race for the second year in a row. He showed his good form in March with a third-place finish in Le Samyn, being beaten only by Philippe Gilbert in the bunch sprint of the peloton, and a few days later he sprinted his way to second place at stage 2 of the Driedaagse van West-Vlaanderen where he was second only to world class sprinter Robbie McEwen. For the third, and last, stage of the race, Breschel would once more sprint against McEwen, with the winner taking the overall victory of the race, this time with the effect that both riders crashed. Breschel broke his vertebrae in two places and McEwen was de-classed in the race.
He came back with thunder and lightning in 2007 and came in an impressive 14th at the Paris–Roubaix, which his team-mate Stuart O'Grady won. After recovering he won his first victory as a professional in stage 2 of Danmark Rundt in August 2007. This was the first Danish stage win in five years of this national tour.
In 2008 his best season came and he got his first big international breakthrough when he on 8 June 2008, won the Philadelphia International Championship also known as the Commerce Bank International Championship in Philadelphia, PA where he outsprinted all contenders in a little bunch sprint after a long and hard race. A couple of weeks later he went on to take another impressive victory when he won the 2nd stage of Ster Elektrotoer, a stage finishing on the feared Cauberg and also won the overall points jersey. He maintained his good form through the season and also came in 2nd in the Danish Road Racing Championship, only beaten by his team made Nicki Sørensen. In August he won two stages at Tour of Denmark and also led the overalls until the final time trial securing him a total fifth place. After all a very impressive season for the young gun the biggest scalp came on 21 September where he won the last stage of the Vuelta a España in Madrid in a very convincing way only a few days after he came in second in the 17th stage of the Vuelta a España. Only a week later Breschel rode very impressively at the world cycling championships finishing 3rd and getting a bronze medal. During the 2010 season Breschel rode well in the Cobbled Classics but suffered from bad luck. Breschel won Dwars door Vlaanderen and put in strong performances in Gent–Wevelgem and Ronde Vlaanderen, but suffered mechanical defects in both races.
He signed for Rabobank for the 2011 and 2012 seasons. In the 2012 Paris–Roubaix he was troubled by a knee injury.
Breschel left Rabobank at the end of the 2012 season, and joined the Danish team Saxo–Tinkoff on a two-year contract from the 2013 season onwards.
Breschel signed for Cannondale–Drapac for the 2016 season.