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.）

verb + て is used to connect 2 verbs. when a verb X is conjugated to 'want to X" form, the い becomes くて and it'll be followed by another verb. the meaning is still the same