To become conceited or vain; to get a swollen head (behave like a 'tengu'—proud/arrogant).
昇進してから彼はすっかり天狗になってしまった。
After his promotion he completely became conceited.
褒められるとすぐ天狗になるタイプだから、注意が必要だ。
He gets conceited as soon as he's praised, so you need to be careful.
成功しても天狗にならず、謙虚さを忘れない方が良い。
Even after success, it's best not to get conceited and to remain humble.