CSV import-export in Python
Rather than going down the OOP route, I would just split this out into a few functions:

import csv

from similarweb import ContentClient

CLIENT_KEY = 'some_key'
FILE_IN = 'similarweb/domains.csv'
FILE_OUT = 'similarweb/new_domains.csv'

def extract_data(path, client):
    """Extract the data from the specified path using the client."""
    lines = []
    with open(path) as read_file:
        for line in csv.reader(read_file):
            url = ', '.join(line)  # see note below
            lines.append(' '.join(
                (url, client.category(url)['Category'])
            ))
    return '\n'.join(lines)

def save_data(path, data):
    """Save the data to the specified path."""
    with open(path, 'w') as csv_write_file:
        csv_write_file.write(data)      

if __name__ == '__main__':
    save_data(FILE_OUT, extract_data(FILE_IN, ContentClient(CLIENT_KEY)))


Per the documentation:


  Each row read from the csv file is returned as a list of strings. No automatic data type conversion is performed. 


so there is no need to explicitly convert to strings.