The homotopy pullback of a point along itself is the loop space I have seen on the nLab that we can view the loop space as a particular homotopy pullback, and that it is even the way a "loop space object" is defined in general (when it exists). 
Can someone give me some intuition of why it is so? 
To which construction in geometry does this correspond (up to homotopy)? 
Any reference would also be welcome.
 A: This can be seen from the following general formula$^1$ for homotopy pullbacks of topological spaces.
Proposition. Let
$$X\xrightarrow{f}Z\xleftarrow{g}Y$$
be a diagram of topological spaces. Then
$$\mathrm{hocolim}\left(X\xrightarrow{f}Z\xleftarrow{g}Y\right)=\frac{X\times Z^{[0,1]}\times Y}{\left((x,\alpha\colon [0,1]\rightarrow Z,y)\ \middle|\ f(x)\sim\alpha(0)\text{ and }g(y)\sim\alpha(1)\right)}.$$
Corollary. Let $(X,x_0)$ be a pointed space. Then the loop space of $X$ is the homotopy pullback of the diagram
$$*\hookrightarrow X\hookleftarrow*,$$
where the inclusion maps send the one-point space $*$ to $x_0$.
Proof. We have
$$\mathrm{hocolim}(*\hookrightarrow X\hookleftarrow*)=\frac{*\times\mathrm{Hom}\left([0,1],X\right)\times *}{\left((*,f\colon[0,1]\rightarrow X,*)\ \middle|\ f(0)\sim f(1)\sim x_0\right)}\cong\frac{\mathcal{L}X}{\left(f\in\mathcal{L}X\ \middle|\ f(0)\sim f(1)\sim x_0\right)}=\Omega X.$$
(Here $\mathcal{L}X$ is the unbased loop space of $X$.)

$^1$See Section 2.1 of these notes by Lennart Meier.
A: Here are some asides on my other answer.
On general loop space objects: See the discussion around (specially above) [Remark 1.1.2.6, HA] and [Section 2.3.2, Mauro Porta's master thesis].

Similarly, the suspension of a topological space $X$ is the homotopy pushout of the diagram:
$$*\hookleftarrow X\hookrightarrow *.$$
What follows is about the unreduced suspension of $X$.
For unpointed topological spaces, we have the following formula$^2$ for homotopy pushouts of topological spaces:
$$\mathrm{hocolim}\left(X\xleftarrow{f}A\xrightarrow{g}Y\right)=\frac{X\coprod(A\times[0,1])\coprod Y}{\left((a,0)\sim f(a)\text{ and }(a,1)\sim g(a))\right)}.$$
Picture from Dugger:

(Unreduced!) suspension of $S^1$ from Wikipedia:


$^2$See Example 2.2 of Dugger's A Primer on Homotopy Colimits.
