There’s nothing self-referential about Tupper’s formula. In fact, Tupper himself didn’t call it self-referential—he merely listed it as a fun example in his paper, Jeff Tupper, “Reliable Two-Dimensional Graphing Methods for Mathematical Formulae with Two Free Variables”, SIGGRAPH 2001 (see Figure 13). Let’s reconstruct it piece by piece to see how it works.

We’ll start by enumerating all the natural numbers in binary: 0, 1, 10, 11, 100, 101, 110, …. We’ll find it convenient to write them “backwards”, with the numbers increasing upwards and the least significant bit on the left:
…
011
101
001
11
01
1
0
We can turn these digits into the graph of a formula. The digit at column [math]i[/math] in row [math]j[/math] should be 1 exactly when [math]\lfloor j 2^{-i}\rfloor[/math] is odd, which is to say, when
[math]\left\lfloor\mathrm{mod}{\left(j 2^{-i}, 2\right)}\right\rfloor = 1[/math].
To make the 1s display as square pixels, note that [math](x, y)[/math] is inside pixel [math](i, j) = (\lfloor x\rfloor, \lfloor y\rfloor)[/math], and substitute that into the formula above:
[math]\left\lfloor\mathrm{mod}{\left(\lfloor y\rfloor 2^{-\lfloor x\rfloor}, 2\right)}\right\rfloor = 1[/math].

By selecting a row of this graph, we can display any one-row binary bitmap image we want. For example, row N = 32753480132533623 = 1110100010111010001110111011100011101010001110101110111₂ of the graph encodes QUORA in Morse code.

We can even write a huge number N for which row N encodes the formula itself in Morse code. You can see where this is going—Tupper’s formula is just a slightly more elaborate version of exactly that idea.

Morse code isn’t very interesting visually, so let’s figure out how to make the graph display images that are more than one pixel tall. We just need to modify the formula to display each natural number in several rows. Let’s start by displaying each natural number …hgfedcba₂ in two rows, with the even-indexed digits in one row and the odd-indexed digits in another:
bdfh…
aceg…
Then the pixel at column [math]i[/math] in row [math]j[/math] represents the [math](2i + \mathrm{mod}(j, 2))[/math]th digit of [math]\left\lfloor\tfrac j2\right\rfloor[/math]. So we just need to replace [math](i, j)[/math] with [math]\left(2i + \mathrm{mod}(j, 2), \tfrac j2\right)[/math] in our formula above:
[math]\left\lfloor\mathrm{mod}{\left(\left\lfloor \frac y2\right\rfloor 2^{-2\lfloor x\rfloor - \mathrm{mod}(\lfloor y\rfloor, 2)}, 2\right)}\right\rfloor = 1[/math].

This lets us display any two-row image. For example, here’s a fun one at N = 16952658865207466 ≤ y < N + 2:

Now we just do the same thing for 17-row images instead of two-row images:
[math]\left\lfloor\mathrm{mod}{\left(\left\lfloor \frac{y}{17}\right\rfloor 2^{-17\lfloor x\rfloor - \mathrm{mod}(\lfloor y\rfloor, 17)}, 2\right)}\right\rfloor = 1[/math].
For some reason Tupper writes [math]\tfrac12 < \lfloor\ldots\rfloor[/math] instead of [math]\lfloor\ldots\rfloor = 1[/math], even though they’re equivalent:
[math]\frac12 < \left\lfloor\mathrm{mod}{\left(\left\lfloor \frac{y}{17}\right\rfloor 2^{-17\lfloor x\rfloor - \mathrm{mod}(\lfloor y\rfloor, 17)}, 2\right)}\right\rfloor[/math].

There we go. That’s all Tupper’s formula is—a gimmick for displaying arbitrary 17-row images, such as, say, this one, at N = 37669073689472413981038757357174421250751615080136175078863731619709820137852535934054444106976288660765441816156055020820524154403449017577048931876895774532717137569451940316881379617664146758321450980308485802669600502037142233219986858109513902092647501593604707677330472943 ≤ y < N + 17:

or this one, at N = 4858450636189713423582095962494202044581400587983244549483093085061934704708809928450644769865524364849997247024915119110411605739177407856919754326571855442057210445735883681829823754139634338225199452191651284348332905131193199953502413758765239264874613394906870130562295813219481113685339535565290850023875092856892694555974281546386510730049106723058933586052544096664351265349363643957125565695936815184334857605266940161251266951421550539554519153785457525756590740540157929001765967965480064427829131488548259914721248506352686630476300 ≤ y < N + 17:

Note that the value N = 960939…404719 from Tupper’s paper is actually wrong, and results in an upside-down formula:

As you can clearly see, there’s nothing self-referential about the formula—all knowledge of the picture is contained in the gigantic number N that defines the specific region the formula is graphed over.

However, by employing the same kinds of techniques used to make a computer program that prints its own source code, it is possible to make a formula that truly contains all the information need to reconstruct its own picture, making it truly self-referential. Some rather long ones were constructed by Tupper himself. Here’s one by Jakub Trávník that’s compact enough to be shown in this answer: