Two answers here present two very different approaches to motivating ideals.
The first presents the historical motivation for ideals, namely Kummer's idea that in some rings like $\mathbb Z[\sqrt{-5}]$ there are somehow "ideal numbers" that allow us to factor products like $2 \cdot 3 = 6 = (1+\sqrt{-5})(1-\sqrt{-5})$ further, and more importantly, uniquely.
The second presents the completely abstract construction of an ideal $I$ of ring $R$ to be exactly a subset that $a \sim b \iff a-b\in I$ is an equivalence relation s.t. $R/{\sim}$ inherits the ring structure of $R$; or as that answer put it, "Equations in $R$ give corresponding equations between equivalence classes in $R/{\sim}$". An answer here phrased this quotient idea as "You can think of ideals as subsets that behave similarly to zero".
Again in Intuition behind "ideal", Qiaochu Yuan says "To me ideals are kernels of ring homomorphisms". This point of view is very connected to the quotient point of view presented above, essentially by the 1st isomorphism theorem. It also makes rigorous the above idea that "ideals are subsets that behave similarly to zero", since kernels are literally the set of elements that get mapped to $0$ by a ring homomorphism.
My question is how to connect this latter, more "abstract" point of view with the historical/Kummer-Dedekind point of view? I don't really have a good intuition/picture for why the set of all elements "divisible by some (ideal) number" should be exactly a subset of elements $S$ s.t. we can "do exactly the same kind of arithmetic" with cosets $\{r+S\}_{r\in R}$ as we can with elements $\{r\}_{r\in R}$.
Best Answer
Your comment that the correct notion of an ideal was not originally designed for forming the quotient of a ring is off the mark. In fact, this was precisely the rationale behind Kummer's introduction of ideal numbers: although there are no prime factors of $3$ in ${\mathbb Z}[\sqrt{-5}]$, Kummer could easily define residue classes modulo "ideal primes"; his main task was showing that these ideal primes could be multiplied in a coherent way. See https://arxiv.org/abs/1108.6066 . If we look at Kummer's theory from our point of view, his ideal numbers are homomorphisms from the ring of integers into finite fields (which he did not know yet, so he had to work his way around them by looking at the decomposition fields); Dedekind's ideals are simply the kernels of Kummer's ring homomorphisms.
Edit. The following is copied more or less from my book on quadratic number fields:
Consider the ring $R = {\mathbb Z}[\sqrt{-5}\,]$. The elements $2$ and $3$ are irreducible in $R$, mbut not prime. If there was an element $\pi$ of norm $2$, then we could consider the residue class ring of $R$ modulo $\pi$; this quotient ring would have two elements, because it can be shown that the number of residue classes modulo an element of $R$ is equal to its norm.
Reduction modulo $\pi$ thus would give us a ring homomorphism $f: R \rightarrow {\mathbb Z}/2{\mathbb Z}$. Kummer realized that such a ring homomorphism exists even when there is no element of norm $2$. In fact, all we have to do is set $f(a+b\sqrt{-5}\,) = a+b+2{\mathbb Z}$. Thus although there is no prime element $\pi$ of norm $2$, we can work modulo $\pi$ by simply applying $f$. Such ring homomorphisms (or, less anachronistically, such procedures for attaching a residue class to each element) were called ideal primes by Kummer.
In the case of ideal primes of norm $3$ there are two ring homomorphisms $\kappa_3$ and $\kappa_3'$ to ${\mathbb Z}/3{\mathbb Z}$, and they are defined by $\kappa_3(a+b\sqrt{-5}\,) = a+b+3{\mathbb Z}$ and $\kappa_3'(a+b\sqrt{-5}\,) = a-b+3{\mathbb Z}$. The kernel of $\kappa_3$ consists of all ${\mathbb Z}$-linear combinations of $1-\sqrt{-5}$ and $3$; but this is the ideal $(1-\sqrt{-5}, 3)$.