How to transfer a domain from 101domain to another registrar?

1. Login at 101domain.com
2. Go to their 'Domains' page
3. Click on the domain in question
4. Buried halfway down that page you'll see a 'Request Authorization Code' link.

Deliberately hard to find if you ask me!

It seems to me that you are not asking to transfer the domain but rather gain access to the dns records.

1. Login as the domain owner and select active domains
2. click on the appropriate domain name
3. on the lower right side of the screen will be the dns servers.
4. Click edit DNS servers and make the desired changes.
5. scroll all the way to the bottom and click next.
6. A new screen will show the proposed DNS changes
7. Click next at the bottom again and your changes will be confirmed.