Describes a gradual change or progression that happens each day. Often used with verbs of change like 変わる, 増える, 成長する.
It's getting warmer day by day.
His Japanese is improving day by day.
Archaic use meaning 'every day' or 'daily'. In modern Japanese, 毎日 or 日々 is used instead.
In old texts, 日に日に can be used to mean 'every day'.
Now that it's autumn, it's getting colder day by day.
The baby is growing bigger day by day.
My grandfather's health is improving day by day.