〜て・〜なくて (reason)
because, and so (cause and effect)
Verb / adjective て-form, result · ない → なくて
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The て-form often carries a reason: 風邪を引いて学校を休みました, "I caught a cold and missed school". The first clause causes the second.
For a negative cause, use なくて: 意味が分からなくて困った, "I didn't understand, and I was stuck".
Adjectives work the same way: 忙しくて時間がなかった, "I was so busy I had no time".
The limit: the result has to be a state, feeling or thing that happened, not a request, an invitation or a decision. 暑くて窓を開けてください is wrong; use から there.
Feelings are the classic case: 会えて嬉しい, "I'm happy to see you (because I can)".
Examples
風邪を引いて、学校を休みました。
I caught a cold and missed school.
皆さんに会えて、嬉しいです。
I'm so glad to meet you all.
意味が分からなくて、困りました。
I didn't understand what it meant, and I was stuck.
雨が降って、試合が中止になりました。
It rained, and the match was called off.
忙しくて、昼ご飯を食べる時間がなかった。
I was so busy I had no time for lunch.