I've opened my Outlook 2010 .PST in Outlook 2013, switched to People then tried to drag and drop my Contacts folders from the Outlook 2010 .PST into Contacts (on this computer) but it doesn't work.

Can anyone advise me how I transfer my Contacts folder from Outlook 2010 to Outlook 2013 please?

Try to export the contacts from the 2010's .pst, and import it to the 2013's .pst.

See

How to export contacts in Outlook 2013:
http://office.microsoft.com/en-us/outlook-help/export-contacts-HA102919759.aspx

How to import contacts in Outlook 2013:
http://office.microsoft.com/en-us/outlook-help/import-contacts-HA102919678.aspx