Suppose $X$ is a smooth variety over a field $k$ of characteristic zero, and $Z$ is a smooth subvariety of codimension d. Now let $\tilde{X}$ be the blow-up of $X$ at $Z$, and let the exceptional divisor be $E$. We have the fibered product, $\require{AMScd}$ \begin{CD} E @>j>> \tilde{X}\\ @V \pi' V V @VV \pi V\\ Z @>>i> X \end{CD}
Let $N_Z$ be the normal bundle of $Z$ in $X$, and let $N_E$ be the normal bundle of $E$ in $\tilde{X}$. From Proposition 6.7 (a) of Fulton's Intersection Theory, the pull back of the cycle (class) Z is given by \begin{equation} \pi^*(i_*[Z])=j_*(c_{d-1}(\pi'^*N_Z/N_E) ~\cap E ) \end{equation} which I do not understand well.
Question could anyone give a more careful explanation of this formula? For example, when $X$ is a threefold, and $Z$ is a smooth curve of codimension 2, $E$ will be a $\mathbb{P}^1$ bundle over $Z$, in this case the formula is \begin{equation} \pi^*(i_*[Z])=j_*(c_{1}(\pi'^*N_Z/N_E) ~\cap E ) \end{equation} i.e. the pull-back is represented by the first Chern class of the line bundle $\pi'^*N_Z/N_E$, is there any geometric interpretation for this case?
