Why Prove You're an American to Vote in the U.S.?

[Link in title]
By SHANNON McCAFFREY, The Associated Press
"The Justice Department has rejected Georgia's system of using Social Security numbers and driver's license data to check whether prospective voters are citizens, a process that was a subject of a federal lawsuit in the weeks leading up to November's election."
...
"In a letter released on Monday, the Justice Department said the state's voter verification program is frequently inaccurate and has a "discriminatory effect" on minority voters..."This flawed system frequently subjects a disproportionate number of African-American, Asian and/or Hispanic voters to additional, and more importantly, erroneous burdens on the right to register to vote," Loretta King, acting assistant attorney general of the Justice Department's civil rights division, said."

First off, according to information I found [link], in Georgia there were 50,000 "voters" that were flagged due to citizenship questions. Of the 50,000 that were purged, 4,500 have yet been able to prove their citizenship.

Second, what exactly is wrong with having to verify you are indeed a citizen before you cast a vote in an election? Put it this way; A few weeks ago my friends co-op was holding a building election. This was an important election, up for grabs this year was both the HOA President and Treasury. As a non-resident in the building, I can't just vote in their HOA election. For obvious reasons I should not be able to vote in the HOA election. Though my single vote could sway the outcome, there is no reason for me to be involved in the politics of this co-op. If the implications of a non-resident voting in an HOA election could lead to long term harm in a co-op, what could illegal votes in a federal or local government election lead to?