につれて, or に連（つ）れて when using kanji, is an expression that may be translated as ‘as (A), then (B)’, or ‘in proportion to (A), (B)’. It is formed through the use of the 格助詞（かくじょし） (case marking particle) に, the る - Verb 連（つ）れる ‘to take along’, and the 接続助詞（せつぞくじょし） (conjunction particle) て. Literally, the translation is similar to ‘being taken along with (A), (B)’, or ‘in tow with (A), (B)’.
につれて will be used after the plain, non-past form of verbs, or nouns.

• 毎日（まいにち）運転（うんてん）をするにつれて、どんどん運転（うんてん）が嫌（きら）いになってきた。
  Along with driving everyday, I am starting to dislike it more and more.
• 夏（なつ）になるにつれて、日（ひ）が昇（のぼ）るのが早（はや）くなってきた。
  Along with it becoming summer, the time the sun comes up is getting earlier.
• 犬（いぬ）の成長（せいちょう）につれて、食欲（しょくよく）が変（か）わる。
  As the dog grows, how much he eats changes.
• 時代（じだい）の変化（へんか）につれて、食文化（しょくぶんか）も変（か）わる。
  As the times change, food culture also changes.

As 連（つ）れる is a transitive verb, (A) につれて (B) implies that (A) is causing (B) to happen simultaneously. This does not have to be something that (A) is purposely doing (although it may be), just a ‘domino’ type effect, where (A) is considered the instigator.

• 冬（ふゆ）になるにつれて、太陽（たいよう）が沈（しず）む時間（じかん）が早（はや）くなる。
  Along with it becoming winter, the time the sun sets gets earlier. (Natural, as winter directly affects the time the sun sets)
• 冬（ふゆ）になるにつれて、犬（いぬ）が成長（せいちょう）していく。
  In tow with it becoming winter, the dog grows more. (Unnatural, as there is no relationship)