We use the form with the auxiliary to add emphasis to the statement, in order to make your point more forcefully.
In the first example, you can imagine that the other person is angry with you for not getting in contact. You want to defend yourself. Maybe you feel that it's not your fault - perhaps you rang your friend's number but her phone was switched off. So you would say 'I did try to phone you' to emphasise the point that you made an effort - it wasn't as though you forgot to contact her. You want to make it very clear that you 'did try'. This is more forceful than the simple neutral statement 'I tried'.