Slowing down the video, he was inside the lane, on the last two steps he stepped slightly outside the lane. But if you freeze the frame at 0:33 and toggle it slowly to 0:34, he beat the ball to the base, and it did not impact the play either at first or at home.
It looks like there was one out. Call the runner out at first, but why take the run off the board from third base? The first baseman was not going to catch that ball.
Could not find a rule that applied here, unless these are not the correct NCAA rules?
EDIT-
I guess this answers it, but still seems like a crap outcome in this particular instance. Life ain't fair sometimes: