会いたくて

want to meet and..., wanting to meet

→ 会う
会い meet (someone) + たく Desiderative (want to) + て Te-form (conjunctive form)
verb · informal Rare #32,707
want to meet and..., wanting to meet The -て form of 会いたい; often links a desire to what happens because of it.
JA: 会いたくて、ここまで来ました。
EN: I wanted to see you, so I came all this way.
Conjugation of 会う

母に会いたくてたまらない。

I miss my mother so much.

彼は早く娘に会いたくてしかたなかった。

He couldn't wait to see his daughter.

彼女にもう一度会いたくてしかたがない。

I can't stop wanting to see her again.