How to remove duplicates, which are generated with array_agg postgres function

You can use the distinct keyword inside array_agg:

SELECT ARRAY_TO_STRING(ARRAY_AGG(DISTINCT CONCAT(u.firstname, ' ', u.lastname)), ', ')
FROM log_has_item logitem
  INNER JOIN log log ON log.id = logitem.log_id
  INNER JOIN worker u ON log.worker_id = u.id
WHERE logitem.company_id = 1

SQLFiddle with this example