First, I don't agree with that interpretation personally (if that's what Chopin had wanted, he would have written it that way, meticulous soul that he was), so I wouldn't put too much stock in how something "should" be played. In this etude, the voices are this flying passage and the melody in the bass. Breaking it up further is artificial-sounding. That's my opinion, and it's no better or worse than the writer's. In other words, what you SHOULD do is make up your own mind. :)
As for what a "voice" is on the piano: voices are just different lines of music played together. This is rigorous in, say, a fugue; a fugue has three or four voices (usually three or four, but sometimes more; five voices isn't all that uncommon) that all run along simultaneously. This version of Bach's first two-part invention (sort of like a fugue with two voices) will illustrate the concept clearly:
Now, once you get past the baroque period, the idea of voices is still there, but more loosely applied. Here's Brahms' Intermezzo Op. 118 No. 2. Have a look at the F# minor section at 1:50. You will see that there are three distinct voices, three distinct musical parts, going on simultaneously. You will also see that Brahms went to considerable trouble to notate the middle (or "inner") voice in such a way that it didn't get lost in the lower voice, even though the lower voice has a lot of the same notes. That's why a lot of the notes have both upward quarter note stems and downward eighth note stems.
Contrast that with the più lento section that begins at 2:29. There's an obvious melody in the top note of each chord, and the rest of the notes are obviously harmonizing chords, so the obvious way to play it (which is not to say the only way to play it) is to bring out the top note in each chord, and let the other notes enrich the melody as a harmonic accompaniment.
So, the composer will give you clues about his intentions with bringing out inner voices, and Brahms was particularly meticulous about this as was Chopin. But also, what you are "supposed" to do is interpret the composer. That means that your performance is a partnership between you and the composer, not a rendering of the music as the composer would play it. You can look for less-than-obvious melodic lines in music, and if you think they express your musical feelings, then that's what you play.
We can go back to Chopin to illustrate this. Here is a performance of his Nocturne No. 9 in Bb. Look at how Rubinstein treats the inner voices; for the most part they are subjugated to the main melody. When he does bring out an inner voice, it's only in small areas, and when it was written that way. An example is at 1:18, the measure just past the one marked "D" in the score.
Now, here's another performance, with an animated score. (I wouldn't personally separate the music this rigorously into voices, but it's one of the requirements of an animated score. Some of the different-colored voices are meant to run together.) You'll notice that Malinowski puts much more emphasis on the inner and bass voices than Rubinstein does (and interestingly, he downplays the inner voice at the point I mentioned in the Rubinstein performance).
So, as you can see, voicing, like phrasing and to a lesser extent dynamics, is very much open to personal interpretation. If someone tells you that you are "supposed" to do it in a particular way, I would take that with a grain of salt. Guidelines rather than rules.