We say that $\tau$ is finer than $\tau'$ if every open set of of $\tau'$ is open in $\tau$.
The question, if so, simply asks to show that every open set of $X$ is open in $X'$. Of course it is enough to show this for basic open sets, in our case $B(x,\epsilon)=\{y\mid d(x,y)<\epsilon\}$ sort of sets.
Fix $x\in X'$ and use the assumption that $f(y)=d(x,y)$ is a continuous function on $X'$ to deduce that $B(x,\epsilon)$ is open in $X'$.
To show that the $d$-metric topology is coarser than $\mathcal{T}$ we must show that every $d$-open set is $\mathcal{T}$-open. Of course, it really suffices to show that every $d$-ball is $\mathcal{T}$-open (since the $d$-balls form a basis for the $d$-metric topology). To show that the $d$-ball $B ( x , \epsilon )$ is $\mathcal{T}$-open it suffices to find for each $y \in B ( x , \epsilon )$ a $\mathcal{T}$-open neighbourhood $V$ of $y$ such that $V \subseteq B ( x , \epsilon )$.
First, the continuity of $d$ implies that for each $\epsilon > 0$ the set $$U_\epsilon := \{ ( u , v ) \in X \times X : d ( u , v ) < \epsilon \} = d^{-1} [ ( -\infty , \epsilon ) ]$$ is open in $X \times X$ with respect to the topology $\mathcal{T}$ (or $\mathcal{T} \times \mathcal{T}$, if you will).
Now, given $x \in X$ and $\epsilon > 0$, we want to show for all $y \in B ( x , \epsilon )$ that there is a $\mathcal{T}$-open set $V$ with $y \in V \subseteq B ( x , \epsilon )$. As $y \in B ( x , \epsilon )$, it follows that $( x , y ) \in U_\epsilon$, and since $U_\epsilon$ is a $( \mathcal{T} \times \mathcal{T} )$-open subset of $X \times X$, there are $\mathcal{T}$-open $U , V \subseteq X$ such that $$( x , y ) \in U \times V \subseteq U_\epsilon.$$ In particular $V$ is a $\mathcal{T}$-open neighbourhood of $y$.
Note that given any $z \in V$ we have that $( x , z ) \in U \times V \subseteq U_\epsilon$, and so by definition of $U_\epsilon$ it follows that $d ( x , z ) < \epsilon$, meaning that $z \in B ( x , \epsilon )$. We may then conclude that $V \subseteq B ( x , \epsilon )$.
Best Answer
Outline: Since $d:X'\times X'\to\Bbb R$ is continuous, then for any $x\in X',$ the function $f:X'\to\Bbb R$ given by $f(y)=d(x,y)$ is continuous. It follows that all the open $d$-balls are open in the topology on $X'.$ Since the open $d$-balls form a basis for the topology on $X,$ then the topology on $X'$ is finer.