$\newcommand{\I}{\mathcal{I}}$ Let $X$ a variety smooth over the complex numbers. Then we know that $\Omega_{X/\mathbb{C}}$ is the (usual) pullback of the conormal sheaf $\I/\I^2$ where $\I$ the sheaf of ideals of the diagonal in $X\times_\mathbb{C} X$. You can of course prove this directly.

One could expect something of this nature (taking $\Omega_X$ to be defined only as the sheaf of Kahler differentials) from computations with the conormal exact sequence

$$ 0 \rightarrow \I /\I^2 \rightarrow \Omega_{X\times X} \otimes \mathcal{O}_\Delta \rightarrow \Omega_\Delta \rightarrow 0$$

$\textbf{Question:}$ Are there geometric ways to see this also? Is there a picture that geometers keep in mind making it obvious that say the tangent bundle to $X$ and the normal bundle of the diagonal $\Delta \subset X\times X$ are really the same thing? This is a very basic concept, yet I can't find such an explanation. For example, is this clear in the context of differential geometry or complex manifolds?