Why do americans say "Nice to meet you" just when they meet someone, shouldn't it be said while ending the conversation?

If the phrase used started with it is or it was, then when it was said would make a difference. But since the exact phrase "nice to meet you" does not include a verbal tense, it can be said at either the start or the end of a conversation.

At the start of a conversation

• ✔ ^{It is} nice to meet you.
  
  Whether the meeting is thought to be a 5-second greeting and handshake or an extended conversation, its still in the immediate present or just-completed past, so is is fine.

• ✘ ^{It was} nice to meet you.
  
  Even if meet is considered to be that 5-second greeting and handshake which has just finished, the past tense would be odd at this point.

At the end of a conversation

• ✘ ^{It is} nice to meet you.
  
  Nobody would normally say this at the end of a conversation. However, it's still less strange than saying it was as the start of a conversation.
  
  
• ✔ ^{It was} nice to meet you.
  
  This is perfectly normal at the end of a conversation.

Without either it is or it was, "nice to meet you" can be said at either the start or the end of a conversation (assuming that the person being talked to has been met for the first time) and be meaningful at either time. In fact, it could be said at both times—for instance, once after shaking hands, and again when waving goodbye.