This is an example from Algebraic Topology, by Hatcher.
As far as I understand, I have to take the direct sum of all the $G_i$s (in this case, $\mathbb{Z}\oplus\mathbb{Z}\oplus...$) and quotient out all elements of the form $(g_1,g_2-\alpha_1(g_1),...)$ with $\alpha_i$ as given in the definition on the same page.
In this case, what are the elements I need to quotient out? Won't they be elements of the form $(z_1,z_2-pz_1,...)$? Because given that $z_2$ was obtained by taking an integer and multiplying it by $p$, $z_2-pz_1$ will be a multiple of $p$.
I just don't see how setting these to $0$ is the same as $\mathbb{Z}[1/p]$.
There's a better way to see why $\Bbb Z[1/p]$ is the direct limit (in my opinion): set up an isomorphism from the original direct limit to another one that's easier to understand:
$$\require{AMScd} \begin{CD} \mathbb{Z} @>{p}>> \mathbb{Z} @>{p}>> \mathbb{Z} @>{p}>> \mathbb{Z} @>{p}>>\cdots \\ @VV{1}V @VV{1/p}V @VV{1/p^2}V @VV{1/p^3}V \\ \mathbb{Z} @>{}>> \frac{1}{p}\mathbb{Z} @>{}>> \frac{1}{p^2}\mathbb{Z} @>{}>> \frac{1}{p^3}\mathbb{Z} @>{}>> \cdots \end{CD}$$
The unlabelled arrows in the bottom row are simple inclusions. The other arrows are multiplication maps and are labelled by the scaling factor. In other words, the map $\mathbb{Z}\xrightarrow{p}\mathbb{Z}$ looks the exact same as the inclusion $\mathbb{Z}\hookrightarrow\frac{1}{p}\mathbb{Z}$, and similarly for all the other parts of the commutative diagram. The direct limit in the bottom is then $\bigcup \frac{1}{p^n}\mathbb{Z}=\mathbb{Z}[1/p]$.
I haven't seen the construction you're talking about, but it seems intuitively clear why it should also define the direct limit. Say we have $G_1\to G_2\to G_3\to\cdots$. We want to interpret the arrows as "inclusions" (any noninjectivity means we pretend elements were the same to begin with in order to maintain this interpretation), in which case the direct limit is the "union" of all the $G_i$s. So our direct limit needs elements from all the things, so we can start off with $\bigoplus_i G_i$ and identify things that are supposed to be equal by quotienting by their differences.
In particular, $g\in G_i$ should represent the same element in the direct limit as $\alpha_i(g)\in G_{i+1}$ (where $\alpha_i:G_i\to G_{i+1}$), so we want their difference $(\cdots,0, g,-\alpha_i(g),0,\cdots)$ within the direct sum to be zero, and thus we must quotient by the subobject comprised of elements of the form
$$(g_1,g_2-\alpha_1(g_1),g_3-\alpha_2(g_2),\cdots)$$
(where of course all but finitely many coordinates are zero). Indeed, the kernel of the homomorphism $\bigoplus_i\mathbb{Z}\to\mathbb{Z}[1/p]$ where the $i$th factor gets multiplied by $\frac{1}{p^i}$ and included is generated by things that looks like $(z_1,z_2-pz_1,\cdots,z_n-pz_{n-1},-pz_n,0,\cdots)$.