What does Whitman mean when he says "What is more subtle than this which ties me to the man or woman that looks in my face? Which fuses me into you now, and pours my meaning into you?"

Looking at this passage, it's hard to decide where to start. What direction should we approach it from to best lead us to understanding?
Literature in general, but poetry especially, can be intimidating to analyze. It seems so elevated and intricate that we think we're going to need super-deep analytical ability to get anywhere near understanding.
I don't think that's true. I think that, almost always, it's best to trust your instincts. You read something, and maybe a word or two sticks out. Hold onto that. Maybe you don't understand a phrase, but you get the general sense what it's talking about. Hold onto that. Those are the ways you're going to approach your analysis: build on what you understand most.
My favorite English teacher once told me that analysis comes down to two things: finding patterns and finding anomalies. Why is the author repeating something, why is the author changing things up? Once you take the things you do understand and look for where they are involved in patterns and anomalies, you'll find connections to the parts you don't understand. Slowly, things start to come into focus. Trust me on this.
I'll show you the way I apply that strategy to this passage. I hope it's helpful, but remember—no two minds work alike. This is far from the only way to analyze poetry.
These were my first three questions: Why did the speaker use the word "subtle"? What is it that ties people together? What is the speaker's "meaning"?
My next question was: Why is the speaker being so vague? It's okay to not worship every word a poet writes. At first glance, this passage seems kind of New Age-y and obtuse, at least to me. But for whatever reason—because it's an assignment in school, because you're genuinely curious, whatever—you've got to analyze it. So keep that thought in your mind—this seems like a silly line—but do the work to see if there's anything more to it.
Of those three questions, "subtle" seems like the easiest one to answer, so I'll hit it first. As I look for definitions of "subtle," I see words like "delicate," "precise," "difficult to describe," "understated," "complex."
Well, those make some sense. "This which ties me" to other people is invisible, so it's definitely understated. It also seems pretty hard to describe. In fact, that's what leads me to my first realization: the speaker agrees with me. He thinks the bond is hard to describe too. That's why he needs a whole 146 lines to do it.
That leads me to look for other places in the poem where the speaker seems to be defining the bond. Line 7 stands out: "The simple, compact, well-join’d scheme—myself disintegrated, every one disintegrated, yet part of the scheme." So there's a large scheme that consists of disintegrated people. I imagine all the people on the ferry melting and running together in a big puddle. The speaker thinks that, on some level, everyone is part of one big thing, the way there are separate water molecules that are all part of one puddle.
Looking for more of the speaker's definitions, I find this: "It avails not, neither time or place—distance avails not; I am with you" (lines 20–21). So the bond is also across time and space. That seems to reinforce our idea that everyone is part of one big thing.
Paydirt! Check this out:

Each came, or comes, or shall come, from its due emission,
From the general centre of all, and forming a part of all;
Everything indicates—the smallest does, and the largest does;
A necessary film envelopes all, and envelopes the Soul for a proper time. (97–100)

This seems like as good a definition as we're going to get. The speaker says each human is emitted from something that forms "a part of all." Maybe the whole world was originally some undifferentiated ball of goo, and everything got shaped out of it, like a bit of clay taken off the pile and sculpted into something.
But there's also the film that envelopes the Soul. That comes up elsewhere, which means we need to pay attention to it. It takes us to line 131: "You necessary film, continue to envelop the soul." This comes in a stanza where the speaker is using the imperative voice (ordering things to do what he says). So let's look around 131 to gather some clues about why the speaker mentions the film here. He's talking about individualization. Line 135: "Keep your places, objects." So we're not all part of one big mush anymore. We each have a film that encloses the stuff that gives us our particular attributes—that makes us different from the world around us.
BUT, that means we were all at some point united completely. In the mush.
The word "film" trips a trigger in my brain. A film is flimsy, thin, light, easily broken. In fact, you might say it's "delicate" and "understated." Subtle!
So there's an anomaly. The lines we're analyzing say the bond tying us together is subtle, but these lines suggest that the film separating us is subtle.
What gives? Then I remember part of an earlier line that didn't really click. Line 100: "A necessary film envelopes all, and envelopes the Soul for a proper time."
Aha! So the film wraps around each of us to make us individuals, but it's also what wraps around the big mush of everything. In other words, we're all using part of the same film. Maybe that's how we're tied together?
But we need to back up for a moment. What on Earth is this guy talking about? The films and the general center of everything and whatnot are not scientific concepts; he must be using them metaphorically. So what represents what? The bits of mush are our metaphorical selves, surely—our souls, maybe. The film is the outside. Maybe he's talking about our exteriors—our bodies, the things that other people can see and touch? The vessels that hold our souls?
Looking back at the lines about films (100 and 131), I notice an interesting pattern. They are surrounded by physical descriptions. After line 100 comes this: "Now I am curious what sight can ever be more stately and admirable to me than my mast-hemm’d Manhattan." Line 130 says, "Appearances, now or henceforth, indicate what you are," and line 132 says, "About my body for me, and your body for you, be hung our divinest aromas." Sight and smell—external things. I think we're onto something here. The films might be our outside selves.
That leads us nicely into the third question. What is the speaker's "meaning"? My first guess is that it is his identity. And scanning the poem, I see he's dedicated a lot of time to describing what exactly makes him who he is. Take this, for example: "I was Manhattanese" (81). Straightforward, sure, but it's part of the way he defines himself—his identity is, in part, the place that he lives. And he spends so much time describing the sights and sounds and smells of Manhattan, I think it's fair to generalize and say that he feels defined by the things around him that he sees and hears and smells and touches.
Something he says early seems to support that: "The impalpable sustenance of me from all things, at all hours of the day" (6). He is sustained by things, always. And something he says at the end all but confirms it:

We realize the soul only by you, you faithful solids and fluids;
Through you color, form, location, sublimity, ideality;
Through you every proof, comparison, and all the suggestions and determinations of ourselves. (137–139)

There we go. The physical has a profound role in shaping the soul.
So where does that leave us? Let's see ... we think the speaker is saying that all human souls come from the same place (the mush). They are made into individuals because there's a film wrapped around them. We think the film represents the outside parts of a human—face and skin and organs and whatnot. Every individual's film is part of the big film that wraps around the mush. That means all of our films (all of our exteriors) are all small parts of the same thing, the way each slice of bread was once part of a loaf.
We know the speaker says that both the film and the soul are shaped by the world around them. Since the speaker focuses so much on what he's seeing and hearing from the ferry, it's fair to assume that he's thinking that these things define him (at least in part).
And from there, we realize that everyone on the ferry is seeing and hearing the exact same things—they share the same environment. It's just one more factor that makes humans similar. We all live in the same world.
Perhaps it's time for the final step: to actually answer the question about what these lines mean. Just for a refresher, here they are:

What is more subtle than this which ties me to the woman or man that looks in my face,
Which fuses me into you now, and pours my meaning into you. (105–106)

The whole stanza is a set of rhetorical questions; the answer seems to be "nothing." Nothing is more stately and admirable, no Gods can exceed these (101, 104). Nothing, therefore, is more subtle than that which ties me and fuses me and pours me.
We have a good idea what it is that ties the speaker to the woman or man, what fuses him into other people—he and they are all made of the same stuff, and even the things that differentiate them slightly (their exteriors) are united by living in the same environment. The unity between people enables them to understand each other on a deep level, if only they realized it.
That's where the subtle part comes in. It has taken the speaker a long time to figure this out, and he has trouble putting it into words. The bond is subtle indeed—hard to pin down, hard to see, hard to articulate. Complex.
In fact, perhaps that subtlety is exactly why the speaker is saying all this. He wants to make it less subtle. He wants everyone to know what he has figured out. He wants to pour his meaning into them.