$\mathcal{W}_2(\mathbb{R})$ is isometric to a convex subset of a Hilbert space (and embedding a hyperbolic space into Euclidean/Hilbert one has a lot of history). 

The argument can find an argument in S.S. Vallender "Calculation of the Wasserstein distance between probability distributions on the line". I will do an informal sketch. 

I want to see probability measures in $\mathbb{R}$ as piles of sand with sand grains enumerated from left to right and indexed by $[0,1]$. For a measure $\mu$ define function $\Phi_{\mu}:[0,1]\rightarrow \mathbb{R}$ by $\Phi_\mu(t)$ be a position of $t$-th grain in $\mathbb{R}$. I claim that for every $\mu, \nu$
$$\mathcal{W}_2(\mu,\nu) = ||\Phi(\mu)-\Phi(\nu)||_{L_2}.$$   

Before proving this let say how the optimal transport looks on $\mathbb{R}$.

Lemma 1: for the measures $\mu$ and $\nu$ the optimal transport plan between them sends $t$-th sand grain of $\mu$ into $t$-th sand grain of $\nu$.

I give the proof of the lemma for the case when
 $\mu = \frac{1}{n}(\delta(\mu_1) + \dots + \delta(\mu_n)),$ for some real 
$\mu_1 < \dots < \mu_n$ and 
 $\nu = \frac{1}{n}(\delta(\nu_1) + \dots + \delta(\nu_n))$,
for some real 
$\nu_1 < \dots < \nu_n$.
 I need to show that optimal transport moves $\frac{1}{n} \delta(\mu_1)$ into $\frac{1}{n}\delta(\nu_1)$ and then the rest will follow by induction. Okay, lets assume its not and (a part of) $\frac{1}{n} \delta(\mu_1)$ is moved to $\frac{1}{n} \delta(\nu_k)$ and (a part of) $\frac{1}{n}\delta(\mu_j)$ is moved to $\frac{1}{n} \delta(\nu_1)$ for some $j,k > 1$. I claim that if we change our transport by moving (the part of) $\frac{1}{n} \delta(\mu_1)$ into $\frac{1}{n}\delta(\nu_1)$ and (the part of) $\frac{1}{n}\delta(\mu_j)$ into $\frac{1}{n} \delta(\nu_k)$ it will get cheaper. Which follows from the following trivial lemma.

Lemma 2: Suppose that $a,b,c > 0$ then $a^2 + (b + c)^2 < (a + c)^2 + b^2$ iff $b < a$. 

Now it's really easy to proof our main statement
$$\mathcal{W}_2(\mu,\nu) = ||\Phi(\mu)-\Phi(\nu)||_{L_2}.$$  
 for the case when
 $\mu = \frac{1}{n}(\delta(\mu_1) + \dots + \delta(\mu_n)),$ for some real 
$\mu_1 < \dots < \mu_n$ and 
 $\nu = \frac{1}{n}(\delta(\nu_1) + \dots + \delta(\nu_n))$,
for some real 
$\nu_1 < \dots < \nu_n$.
Indeed, squares of both sides are equal to 
$$\frac{1}{n}\sum_{i=1}^{n}|\mu_i - \nu_i|^2.$$

PS: note that measures of the type $\mu = \frac{1}{n}(\delta(\mu_1) + \dots + \delta(\mu_n)),$ for some real 
$\mu_1 < \dots < \mu_n$  are dense in Wasserstein space [see Proposition 2.10 from Karl-Theodor Sturm. "On the geometry of metric measure spaces."][1]. So all the above can be formalized with some suffering.


  [1]: https://scholar.google.com/scholar_url?url=https://projecteuclid.org/journals/acta-mathematica/volume-196/issue-1/On-the-geometry-of-metric-measure-spaces/10.1007/s11511-006-0002-8.pdf&hl=en&sa=T&oi=gsb-gga&ct=res&cd=0&d=861907149611024706&ei=3iekY8OlML2Uy9YP0qeC-A8&scisig=AAGBfm0JbJpdlojQ8qBqaFSCKZHUXOB9MA