transfer of contacts from iPhone 5s to iPhone 6.

Contacts are always part of an email account.

You may have had either iCloud, Google, Yahoo, Outlook, AOL, Exchange or any other email account. Add it in your new iPhone all your contacts will be there too.

See the pic below

The iPhone does not use the SIM to read or write anything except cellular account info. Contacts on the iPhone are managed in several ways, iCloud, another cloud service, email accounts, or just directly on the phone. If you wish to move contacts from one phone to the other, you can either restore the new phone to the backup of the old one if you are just managing on the phone, or you can sign in with the iCloud ID on the new phone and turn on contacts. If you are using an email account to manage them, then you would need to add that email account to the new phone and turn on Contacts.