How does that purification of mind happen? He gives it in the first half.
Svadharma-nishhTA : (he who is) fully and firmly established ( nishhTA) in one’s own dharma.
Tayaiva Atma vishuddhiH : By that alone the mind gets purified Well, how does one know what that svadharma is?
The answer is already there in the very bveginning of the shloka:
Shruti-pramANaika-mateH: The nishhThA of svadharma comes from the unique faith and comviction that the religious sanction comes from only the vedas.
What the Vedas say is the authority. With that faith one goes about doing his svadharma as the be-all and end-all.
That is svadharmaika-nishhThA.