Both are correct. Without "the" is preferable in my opinion under the rule of omit needless words. There is no why. Grammar is arbitrary. It's just the way we do it.

It needs another "in." But it also needs a logic transplant. My call is just one call and therefore does not have an order. Only the group of calls has an order. Therefore, the announcement should say "Calls are being answered in the order they were received in."

It's wrong on 2 counts: (1) My call is just one call. It can't have an order. Only a group of calls has an order. (2) It's missing an "in" at the end. The correct statement would be "Calls are being answered in the order they were received in."