Let's take the definition of $\mathbb{hocolim}$ as the representation of the representable functor like this:
$\underline{\cal M}(\mathbb {hocolim}_{ \cal D} F,m)\cong \mathrm {{sSet}^{\cal D^{op}}}(N(-/\cal D),\underline{\cal M}(F,m)) $,
using weighetd colimit weighted by the nerve of $\cal D$.My question is why this object must be equipped with a universal simplicial natural transformation
$N(d/\cal D)\Rightarrow \underline{\cal {M}}(Fd,\mathbb{hocolim}_{ \cal D} F)$. What information this n.t. gives to us and what it means universal?
For the definition of the data used in this post see Riehl:Categorical Homotopy Theory,CUP 2014.,page 117
The short answer to your question is yoneda lemma.
The longer answer is the following. By yoneda lemma we know that for every co-presheaf $P \colon \mathbf C \to \mathbf {Set}$ and every object $c \in \mathbf C$ there is a bijection $$\mathbf{Set}^C[\mathbf C[c,-],P]\cong P(c)$$ which is natural both in $c$ and $P$ (i.e. it is a natural transformation between the (bi)functors $\mathbf{Set}^\mathbf{C}[y_{-},-]$ and $\text{eval}$, the evaluation functor, objects of the category $\mathbf{Set}^{\mathbf C\times \mathbf{Set}^\mathbf C}$).
This natural isomorphism sends every natural isomorphism between $\mathbf C[c,-]$ and $P$ to an element in $P(c)$ which is universal for $P$ (take a look to any standard book in category theory for the details).
If you want to use the characterization of homotopy colimits as a representable objects you have to remember that in this case a homotopy colimit is not just an object: it is the data given by an object, namely $\text{hocolim} F \in \mathcal M$, and a natural isomorphism between the functors $\mathcal M[\text{hocolim} F,-]$ and $\mathbf{sSet}^{\mathcal D^\text{op}}[N(-,/\mathcal D),\mathcal M[F,-]]$.
By yoneda lemma this natural transformation can encoded as an element of $$\mathbf{sSet}^{\mathcal D^\text{op}}[N(-/\mathcal D),\mathcal M[F,\text{hocolim} F]]$$ that is a natural transformation $N(-/\mathcal D) \Rightarrow \mathcal M[F,\text{hocolim}F]$ and this n.t. allows one to recover completely the isomorphism between the representable functor and the presheaf above, so it completely determinate the representable object, hence the homotopy colimit.
Hope this fully address your doubts. Feel free to ask if you need additional details or clarifications.