につれて

As X happens, So does Y
Plain form verb + につれて
Noun + につれて

につれて is a phrase that expresses a change related to some action.
The change should be in proportion to the action. 

Examples

• 時間がたつにつれて、かれがどんどんいら立った
  As time passed, he got more and more annoyed.
  
  
• 雨が降るにつれて、河の水が高くなる
  As it rains, the water in the river gets higher
  
  
• 年を取るにつれて体がだるくなる
  As you get older, your body becomes slower. 
  
  
• 季節が変わるにつれて寒くなった
  As the season changed, it got colder.
  
  
• 秋になるにつれて紅葉が見えるようになった
  As it became fall, became able to see fall leaves.
  
  
• 大人になるにつれて、一緒に遊ばなくなった
  As they grew up, we played together less and less. 
  
  

See also とともに (Change)

Origin

につれて comes from つれる, a verb meaning something like “pull with”. You’ve probably heard it used as 連れて行く – bring with you.
You can think of につれて as “pulled along by” if it helps.