An audio interface is an external soundcard.
So I'm guessing he meant vs. an internal soundcard, like the one that comes with just about any computer, or like the soundblaster lines, that sort of thing.
Of course, there are audio interfaces that are internal (like the m-audio delta 1010), so it's not really a hard and fast distinction.
But, generally, yes, people agree with this - your final product will sound better, all things being equal, if you invest in higher quality recording gear.