http://www.snopes.com/obama-encouraged-illegal-aliens-to-vote/
Haven't gone into the specifics. Basic jist is that the video was heavily edited, supposedly.
Snopes - typical when it comes to political issues - pulled a fast one here by focusing on the (incorrect) claim that Obama
encouraged illegals to vote, while ignoring the accurate claim that they would not be punoshed if they did.
In some versions, the video was shortened at the end, but the key question and answer was not altered.
The interviewer specifically asked about the consequences if "Dreamers" or someone "undocumented" voted. She also said in her question that she considered undocumented people to be citizens "because they contribute to this country."
After that preface, she asked Obama if they or their familes would be punished if they voted, and Obama flatly said "no". He then stated that voting records are not reviewed by immigration authorities so punishment would be impossible.
At that point, she gets this big smile on her face and is nodding.
After that, he goes into a longer speech about voting. At one point, he does say that only citizens can vote. But of course, she already established her definition of a "citizen", which he did not dispute or clarify.
And most importantly, even if you interpreted his answer as saying that Dreamers and other undocumented people could not legally vote, his statement that they would not be punished if they did still stood. Which is why the interviewer was still smiling.