Bairstow kept wicket for England in the final two Test matches of their doomed tour to Australia earlier in the year, and was left out of the 12-man squad for the opening match against Sri Lanka on Thursday.
The 24-year-old has 14 Test caps to his name but regular keeper Matt Prior has regained his spot while the likes of Moeen Ali and Sam Robson have earned spots in the batting line-up.
"It's disappointing not to get selected but that's the nature of professional sport," Bairstow told Sky Sports News. "I've just got to go out and score as many runs as I can.
"Of course it hurts – everyone wants to play for their country. It's hurting deep down inside.
"It's important for me to get out there, keeping wicket and scoring runs, doing what I've been doing for the last four years."
Bairstow currently averages 41 with the bat in the County Championship Division One, and has claimed seven dismissals behind the stumps.