There are a few ways to answer this, but the spoiler is that it's all about overtones.
For the example from Smells Like Teen Spirit:
"Here we are now" sounds like a chorus effect to me on top of multitracking, which was (and still is) a common way to give vocals a thicker sound.
A chorus typically delays the input sound by an oscillating amount of time (via LFO), then feeds the delayed output sound back at the same time as the input sound. What I think you are hearing there is that the pitch increases slightly when the delay time is decreased and drops slightly as the delay time is increased. This slightly different pitch played on top of the fundamental gives a slightly "thicker" quality to the sound. We are hearing small differences in the fundamental tone AND small differences in the overtones at the same time, which makes the "split" sound more prominent.
However, I don't think all of the "split sound" is due to chorus effects. There are some vocal aspects to that too, like in your demo. So how do you sing so as to intentionally produce that split sound? The answer is overtones.
Without a physics lesson, overtones are are harmonics that also include non-integer multiples of the fundamental. For the sake of understanding, compare the sound of a saxophone to the sound of an ocarina. The ocarina is pretty close to a pure tone instrument (ie, very little audible overtone sound), while the saxophone has a lot of overtones. Organ players might have an even more intuitive grasp on what overtones sound like because stops tend to mess around with the balance of overtones in a particular sound. In general, more overtones means a "grittier" and more complex sound (think saxophone vs ocarina).
Fun fact, the primary thing you hear changing when vowel sounds change is the strength of various overtones (we usually call the peaks in amplitude at particular frequencies "formants"). Tuvan throat singing, technically called "Polyphonic Overtone Singing", is a technique where vocalists have learned to emphasize overtones MORE than the fundamental note (mostly by changing vowel shapes), and it sounds awesome. That would technically be a kind of "split note" as they are literally singing multiple distinctly audible notes at the same time.
Though all that context is relevant, the direct answer to your question "How do I sing a split note" boils down to this: add more overtones to your singing.
There are quite a few ways to do this vocally. I'll outline a few of them:
-You know that angry sort of growly "HUMMMPH!" sound that people make when they are frustrated? This sound often comes out with some emphasized overtones (you'll hear multiple notes at once). When I tried it, my little baby girl just did it next to me and showed me that she could do it too. This is a great starting point to determine what your vocal instrument needs to do to produce a "split sound" as you called it (ie, produce and emphasize more overtones).
-Growling adds overtones to your sound and can give that "split tone" sound. There are some good tutorials on growling while singing without hurting yourself by rock stars out there.
-You can play with the balance of overtones by changing vowel sounds, pushing more air without changing throat positions, and by pushing the sound deeper into your chest or further into your nose (which is an intuitive way of talking about changing the shape of your vocal instrument at various points).
-For guys more than gals, one way to learn how this might feel is to find a place where your break to falsetto naturally occurs, then try to sing both sides of the break at the same time. You'll sound like a boy going through puberty, but if you can get them at the same time you'll get a somewhat obnoxious growling harmonic sort of noise. In my experience, some people have a hard time finding this spot, so don't be frustrated if you can't find it right away.
PS -- I'm a male vocalist, so these things might apply a little differently for female vocalists.
