Pattern: XにつれてY. Indicates that Y changes gradually as X changes. Often used with verbs of change like 変わる, 増える, なる.
年を取るにつれて、体力が落ちる。
As one gets older, physical strength declines.
日が経つにつれて、悲しみは薄れていった。
As the days passed, the sadness faded.
につれて and に従って both mean 'as X changes, Y changes accordingly', and are largely interchangeable, but につれて feels more natural in casual speech while に従って can sound slightly more formal or written.
From the particle に and the te-form of the verb 連れる (to take along, to lead), literally 'being taken along with'. The meaning shifted to indicate a proportional or accompanying change.