You can consider ～くて as te-form, which connect two sentences together. The tense is decided by the last verb.
あなたに会いたい。I want to see you. (Sentence finishes here.)
あなたに会いたくて、ここに来た。I wanted to see you(sentence continues), so I came here.
Sometimes, you may omit the second sentence and leave the rest to the implication.
あなたに会いたくて・・・（so I came.）