In a statement, the 23-year-old revealed his mother is suffering from a "serious, incurable illness".
"In the run-up to the match against Burnley, I felt completely frazzled and unable to concentrate properly," said the centre-back.
"I thought that my distraction may result in an error."
He added: "After much thought and consultation with the gaffer, my family and my team-mates, I felt unable to guarantee my usual standard of performance and said as much to the gaffer."
Tarkowski was originally named in the starting XI to face Burnley in the Championship, but was subsequently omitted from the matchday squad as the Clarets beat Brentford 3-1.
The former Oldham player was disciplined by the west London club and has since returned to training, but was left out of the squad for Saturday's trip to Preston and will not feature against Leeds on Tuesday.
"I would like to apologise to my team-mates, the gaffer and the fans," said Tarkowski.
"I hope you can understand the pressure I was under and that no offence or insult was intended to anyone associated with the club.
"I have taken the sanction given to me by the club with good grace."
Brentford have rejected transfer bids for Tarkowski this month, saying offers have not met their valuation of the defender.
Bees head coach Dean Smith says he has been "reintegrated" into the squad, but the situation is "too raw" for Tarkowski to return to action immediately.
"We will try to help him through this tough period he has had," said Smith.
Brentford defender James Tarkowski says personal reasons were behind his request not to play for the Bees against Burnley earlier this month.