Well, such a form has several meanings depending on the case. I am going to provide a brief overview, but it can be helpful. The pattern is always the same:
(Verb:ない-Form:Drop->い)くて、(Sentence)
I am going to provide a direct answer for the specific case pointed out in the question, then, considering that you wanted a generic explanation about the 〜くて form, I am going to post more cases.
In your specific case...
In your case you have a sentence where the 〜たい form is applied. Such a form has the following pattern:
(V:ます-Form:Drop->ます)たい
And is used to express will, intention. For example:
行く ==> 行きます ==> 行きたい
Please note that the 〜たい form turns your verb into an い-Adjective. After this, you apply the 〜くて pattern to the verb/adjective and get the 〜くて form. So what is the meaning? Among the possible ones, one is the one reported below.
〜くて as a conjunction
When used for い-Adjectives, the following pattern:
(い-Adjective:Drop->い)くて、(Sentence)
Can be used to put together some sentences. For example:
あのね、石川先生は面白くて、優しくて、ハンサムな人ですよ! ==> Well, Ishikawa-sensei is interesting, nice and a beautiful person.
You can make concatenation of adjectives using this pattern.
About your answer...
last option I showed in this post (some sort of ellipsis as someone else described before):
会いたくて ==> I want to meet (him/her?)... ==> I want to meet him/her but... ==> I would like to meet him/her, but...
For a better comprehension, post the full sentence.
Other meanings associated to the 〜くて form...
I promised since you asked for a generic answer. Below I listed some other meanings of the 〜くて form.
In spoken Japanese, used to leave a certain "negation"-like feeling in the air
It is used very often in spoken language and in conversations. Consider the following example where 笑{えみ}子{こ}, a woman working in a shop, is apologizing because she picked the wrong color for a t-shirt a client wanted to check:
笑{えみ}子{こ}さん: あー、ごめんなさい、やはり色が違いますね? ==> Oh, I am sorry, as I thought the color is wrong right?
お客様: じゃなくて、色大丈夫です、サイズは小さすぎるんでしょう! ==> Well, not really!
The problem is the size, it is too small!
笑{えみ}子{こ}さん: そうですか、解りました!ちょっと待って下さい、すぐ見に行きます。Ah! You are right! Please wait a moment, I am going to check!
As you can see, Emiko thought one thing, but the problem turned out being a completely different one.
Consider this different situation:
お父さん: じゃ、これですね?カバンを買いに行きました。無くしたので、カバンが要ったんじゃない〜 ==> Well, Is this it? I went buying a new bag for you, since you lost yours. You needed it right?
息子さん: えと、要らなくて〜カバンを無くしたけど、そのあと友達に見つけてもらったんですから〜 ==> Well, don't really. I didn't lost it, a friend found it for me, so...
Daddy thought his son needed a new bag, but the son told him that it was not necessary.
In these kinds of situation, when you want to express a little disappointment and embarass for something not necessary, you use this form.
To express the cause for something that happened
You can use the same pattern in order to point out the cause for a certain event to happen. However this form can only be used for verbs in negative forms. Pattern is:
(Verb:ない-Form:Drop->い)くて、(Sentence)
Consider the following example:
遅くて、ごめんなさい。 ==> Sorry, I was late
The previous is not a sentence really used a lot in spoken language, to say sorry for being late Japanese use other forms, but sometimes kids use that. Consider this example as well:
何も無くて、よかったね! ==> Thank god nothing happened!
Or this:
課長が何も言わなくて、こわい! ==> Boss didn't say a single word, he was scary!
To express something like: "no need of..."
The pattern is the following:
(Verb:ない-Form:Drop->い)くても、(Sentence)
But Japanese people, only in common conversations and where all people are at the same level (so no boss, no section manager and so on...), do tend to drop も from 〜くても, so the pattern can also be:
(Verb:ない-Form:Drop->い)くて[も]、(Sentence)
Where も in square brackets means: optional. The meaning of this pattern is: "No need of...". Consider the following example.
パーティーへ行かなくても、大丈夫ですよ! ==> Even if I do not go to the party, everything will be all right.
プレゼントをもらわなくても、うれしいです。 ==> Even if I did not receive a present, I would be happy.
When the sentence is "いいです", so we have the following pattern:
(Verb:ない-Form:Drop->い)くて[も]、いい[です]
The meaning is: "No need...". Consider the following example:
明日は会社に来なくてもいい! ==> Tomorrow you do not need to come to work.
Or this example:
払わなくて、いいですよ! ==> You don't need to pay! ==> It is ok if you do not pay.
Or this as well:
見つけなくても、いいですよ。大切な物じゃないから〜 ==> It's all right if you don't find it, it is not something important (referring to something lost).