The above picture comes from Fulton's "Introduction to Intersection Theory in Algebraic Geometry". The variety $I$ is the partial flag variety $F(0,d;n)$, also known as the incidence variety of points on $d$-planes in $\mathbf P^n$. In other words, points in $I$ are pairs $(p,P)$ where $P$ is a $d$-plane and $p$ is a point on $P$. We assume $0<d<n$.
The Chow ring of $I$ has a basis given by classes of the cycles of Schubert varieties, or simply Schubert classes. We can concretely label these classes by $(a_0,\dots,a_d; k)$, where $a_0,\dots,a_d$ is a strictly increasing sequence of integers in $\{0,\dots,n\}$, and $k$ is an integer in $\{0,\dots,d\}$. Fulton uses the notation $(a_0,\dots,\overset{*}{a_k},\dots,a_d)$. Ignore his use of the notation $\{1,\dots,\overset{*}{1},0,\dots,0\}$. The dimension of such a class is $k+\sum\limits_{i=0}^d (a_i-i)$.
Since $V'$ has codimension $d+1$, the class of $[V']$ is a $\mathbb Z$-linear combination of Schubert classes of codimension $d+1$. It is clear to me that all of the $\mu_k$ have codimension $d+1$. However, it is not so clear why there should be no other Schubert classes of codimension $d+1$ appearing in the expansion of $[V']$.
First, it is not the case that the $\mu_k$ are the only Schubert classes of codimension $d+1$. Indeed, assuming that $d+2\leq n$, we have $(n-d-2, n-d+1, \dots, n; 1)$ has codimension $d+1$.
Then, in order to verify that the other codimension $d+1$ Schubert classes don't appear in $[V']$, one would have to understand what all possible codimension $d+1$ Schubert classes are and how they interact with $V'$.
Is this even possible? Am I missing something?