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.
As one gets older, physical strength declines.
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.