The て-form of the negative ない, なくて, can express a reason or cause – meaning "because (something) doesn't happen / didn't happen" or "because there is no…". The negative situation described in the first clause serves as the reason for the result in the second clause.
When なくて expresses a reason, the second clause typically describes an uncontrollable result – an emotion, a state, or a situation that happens naturally. Common results include feelings like かなしい悲しい, こまる困る, しんぱい心配する, ざんねん残念; or involuntary outcomes like できない or おくれる遅れる.
A common point of confusion is the difference between なくて and ないで when giving a reason. Use なくて when the result is an involuntary state or emotion. Use ないで when the result is a deliberate action (meaning "without doing").
Although the structure focuses on verbs, なくて as a reason also works with い-adjectives (おいしくなくて), な-adjectives (静かじゃなくて), and nouns (学生じゃなくて) in the same way.