One of the ways to define a limit of a functor $F:\mathsf C\longrightarrow\mathsf D$ is a representation of $\mathsf{Nat}(\Delta-,F)$.
Along the journey of generalization to the enriched setting, one notes there's a bijection natural in $d$: $$\mathsf{Nat}(\Delta d,F)\cong \mathsf{Nat}(\Delta\mathbf 1,\mathsf D(d,F-)).$$
Thus a limit may be equivalently defined as representing of $\mathsf{Nat}(\Delta\mathbf 1,\mathsf D(-,F-)):\mathsf{D}^\text{op}\longrightarrow \mathsf{Set}$.
Now, the diagram encoded by $\mathsf{D}(d,F-)$ is just the canonical projection $d/F\rightarrow \mathsf C$, which is the base of a cone with vertex $d$ (the vertex is forgotten). Hence, I would think that a natural transformation $\Delta d\Rightarrow F$ should be the same as a natural transformation out of the "one object diagram" at $d$", i.e just $d$, into $\mathsf D(d,F-)$.
However, the diagram encoded by $\Delta\mathbf 1$ is $\mathsf C$ itself, so a natural transformation $\Delta\mathbf 1\Rightarrow\mathsf D(d,F-)$ doesn't graphically seem to give a cone.
What am I missing here?
Be explicit: categorical definitions tend to hide intuition as implementation details of the abstract structure described: it's very easy to get confused regarding what the definitions actually capture (as you do when it comes to interpreting the functor $D(d,F-)$). In particular, when using the correct interpretation of $D(d,F-)$, a natural transformation $\Delta\mathbf 1\Rightarrow D(d,F-)$ does indeed look like a cone.
Explicitly, $D(d,F-)$ takes $X\mapsto \{d\to FX\}$ and $X\xrightarrow{f} Y$ to $\{d\to FX\}\xrightarrow{Ff_*}\{d\to FY\}$; in particular this is certainly not the projection $d/F\to C$, but the fibers of the projection, so it's an "inverse functor" the same way that an inverse function $Y\to X$ gives preimages $f^{-1}(x)=\{y\in Y:f(y)=x\}$.
The relevance of the comma category $d/F$ (and of the "inverse functor" of its projection) to cones and weighted cones is this. The objects of $d/F$, i.e. morphisms $d\to FX$, are the exactly the projection morphisms that cones (and weighted cones) with vertex $d$ are made out of. A cone with vertex $d$ for $F$ is nothing other than a coherent choice of a single object of $d/F$ for each object $X$ (the morphism structure on $d/F$, and hence of $D(d,F-)$ is what allows us to say when such a choice is coherent).
In other words, to give a cone with vertex $d$ over $F$ means that for each object $X$ of $C$ we have to make a choice in $D(d,FX)=\{d\to FX\}$, and these choices have to be coherent. Such choices are made by giving morphisms $\mathbf 1\to D(d,FX)=\{d\to FX\}$ for each object $X$ in $C$, and the coherence of the choice is exactly the statement that these are the components of a natural transformation $\Delta\mathbf 1\Rightarrow D(d,F-)$.
Hence, natural transformations $\Delta\mathbf1\Rightarrow D(d,F-)$ are exactly cones with vertex $d$ over the diagram $F$, assembled by coherently choosing projection morphisms to $FX$ for each $X$ in $C$.
More generally, if you have a functor $C\xrightarrow{W}Set$, a natural transformation $W\Rightarrow D(d,F-)$ is exactly a weighted cone with vertex $d$, that is, a choice for each $X$ of a $W(X)$-(pointwise-)indexed family of morphisms $d\to FX$, coherent in the sense that post-composing with $FX\xrightarrow{Ff}FY$ takes an element indexed by $w_X\in W(X)$ to an element indexed by $Wf(w_X)$ in $W(Y)$. In other words, the weights of weighted cone are distributed on the projection morphisms of the cones.