Anonymity in social welfare function
Anonymous/impartial SWFs focus only on the pattern of well-being,
  and not the identities of the people who end up at particular well-being levels.


Identities here simply means names. When applying an impartial SWF, one only looks at the profile of utilities, not who those utilities are associated with. Consider the following two cases: 
\begin{array}{ccc}
&amp;U_{\text{President}}&amp;U_{\text{CEO}}&amp;U_{\text{Farmer}} \\\hline
\text{Case 1}&amp; 100 &amp; 80 &amp; 10\\
\text{Case 2}&amp; 10 &amp; 100 &amp; 80
\end{array}
The patterns of the distribution of well-being/utility in the two cases are essentially the same, and yet they differ in who is better/worse off. An impartial SWF would generate the same outcome in both cases, since it only focuses on the pattern (i.e. the utility numbers) of the well-being distribution and not on who (i.e. the identity) has which level of utility. In particular, the fact that the President is the happiest person in Case 1 but the most miserable in Case 2 has no impact whatsoever on the outcome of the impartial SWF.