You can be against things such as racism and not be a Marxist/Leftist. In fact pious and highly-theoretical brands of anti-racism actually contribute to racism.
If a Leftist admits to having “white privilege”, then he must feel a certain sense of guilt about having that white privilege. People who are privileged often feel guilt for that privilege; as do the numerous Marxists who have been financially and educationally privileged (as well as being white). In fact their white guilt probably motivates almost their entire position on anti-racism. Guilt is an extremely strong emotion/psychological disposition.
And since nearly all Marxists/Leftists are both white and from well-off backgrounds, that is a double dose of strong guilt and it may well motivate their entire politics.
The very acceptance of one's“privilege” involves guilt otherwise it wouldn't be seen a privilege in the first place.