ジョンは入り口でチケツトを二枚出ました。
 

Setting: John and his friend are going on a date at a Kabuki-theater.

ジョンは入り口でチケツトを二枚出ました。

I'm wondering especially about the meaning of 枚 (mai) in this setting.

Counter. In Japanese you can't use directly numbers, but you need to use them with a counter particle.

So 枚 means counter? I'm not sure if I understood.

Every object, person or animal in Japanese have to have a counter in order to express a quantity. 枚 is used to count flat objects. So if you want to count the tickets, you can't simply say ニチケット, but you need to place a counter. For tickets the counter is 枚, so becomes 二枚のチケット. There are different counters, so for example if you want to say 2 dogs it will be ニ匹の犬 since the counter used for small pets is 匹 (ひき).

It should be 出しました(だしました, transitive verb), not 出ました(でました, intransitive).

枚 means "piece" as in "three pieces of paper" in English. In English, only uncountable nouns have counters. But in Japanese and Chinese, almost all nouns have counters, which is one of the most difficult parts for foreign learners.

I answered it in another thread.
Japanese counter word - Wikipedia, the free encyclopedia

That's a very good catch!

Also, it should be ジョンさん, not ジョン.

Wow! I didn't know that. This should make things easier for me understand in the future. Thank you!

yes there is a counter for everything, but correct me if I'm wrong, you can count (almost) everything with "つ".

一つ
二つ
三つ　
etc.

I think Sashimister posted once before here that using the 〜つ counter so much will make you sound unnatural and uneducated.