I am trying to understand Huybrecht's definition of the blow-up of a complex manifold $X$ along a submanifold $Y$ - if you don't have a hard copy to hand, I have found an electronic version here (see page 99). I understand his explanations of blow-ups in $\mathbb{C}^n$, and I think I follow up until and including the definition of $\hat{\phi}$.
At this point, I am struggling to understand what is being done. Firstly, I am unsure as to what the use of $\hat\phi$ is - are these the so-called gluings at the top of page 100? Are we meant to be taking $U=U_i$ and $V=U_j$ to be charts and $\phi$ to be the transition functions $\phi_{ij}?$ If this is the case, then I don't understand why $\phi_{ij}|_{\mathbb{C}^m}$ are the transition functions for the normal bundle $\mathcal{N}_{Y/X}$. If someone could please clarify this explanation, it would be much appreciated - I am struggling to find another source that defines blow-ups in the same manner as Huybrechts without resorting to algebraic geometry. Thanks!
$\hat{\phi}$ are not the gluings referred to at the top of page 100, but they are used to help produce them.
$U$ and $V$ are not charts on $X$. Instead, they are meant to be like $\varphi^{-1}_i(U_i\cap U_j)$ and $\varphi^{-1}_j(U_i\cap U_j)$, open sets which are identified by the transition function $\phi$ between the two charts $U_i$ and $U_j$. The reason for doing this is that we know that $\sigma^{-1}(U_i)$ and $\sigma^{-1}(U_j)$ are perfectly fine complex manifolds, and our objective is to show that they can be glued - since blowing up was fairly intrinsic to define, our hope is to bootstrap the pre-existing gluing of $U_i$ to $U_j$ which is given by the isomorphism $\phi:U\to V$ to do the correct work. Successfully carrying out this work means producing an isomorphism $\hat{\phi}:\sigma^{-1}(U)\cong\sigma^{-1}(V)$ and showing compatibility over any triple intersection.
Since $\hat{\phi}$ produces such an isomorphism of $\sigma^{-1}(U)$ and $\sigma^{-1}(V)$, it remains to check that it's compatible on a triple intersection (the cocycle condition). On $X\setminus Y$, $\sigma^{-1}(S)$ is just $S$ for any $S$, so we already know compatibility by assumption about the charts. It remains to check how things work on the preimage of $Y$.
The most important thing to note next is that the $\phi_{ij}$ referred to on page 100 are the same $\phi_{j,k}$ referred to in the final paragraph of page 99 - these have nothing to do with the charts we were discussing earlier, and are in fact just the appropriate coordinates that determine what line we end up on in the incidence variety (Proposition 2.4.7 makes this clearer and actually shows that these are literally exactly the transition functions for $\mathcal{N}_{Y/X}$). So we already know they satisfy the cocycle condition and therefoer we're good.