According to the flesh: Paul often uses the word flesh (Greek sarx) to speak about bodily life in this world (for example, 4:1; 8:3).
Paul mentions King David’s family because God promised that a person descended from David would be the Messiah (God's chosen one) and would have an eternal kingdom (2 Samuel 7:13–16; see Isaiah 9:7; Jeremiah 33:15). Jesus was born into David’s family line (Matthew 1:6; Luke 1:27, 32). This shows that he met God’s requirement to fulfill this promise.