This is from Lang's $Algebra$, revised third edition, page 52. I will state my understanding of the problem.
Suppose $G$ is a group, and $F$ is a family of normal subgroups of $G$, partially ordered by the subset relation. Then $F$ is directed, so we define the inverse limit: $\lim_{H∈F}G/H$ by the canonical homomorphisms.
Suppose we have an $A⊆F$ and $A = \{H_i\}_{i∈\mathbb{Z_+}}$, where for all i, we have $H_{i+1}⊆H_i$. Then we also similarly define the inverse limit $\lim_{i∈\mathbb{Z_+}}G/H_i$.
The problem is to show that these two limits are isomorphic if A is cofinal, that is, if for all $H∈F$ there exists $H_i∈A$ such that $H_i⊆H$. The problem seems to be counterintuitive and I do not know how to start. I would be thankful for hints as well as answers.
After user10354138's comment it is an easy task to arrive at the desired isomorphism.
As for the intuition, it follows directly from the proof; the $H$th entry of $x ∈ \lim_{H∈F}G/H$ contains information about all larger $K$ encompassing $H$ and therefore the "collapsed" form of $x$ in the countable limit contains information about the original $x$ that is entirely recoverable.