Let $\mathcal{C}$ be an $\infty$-category, and let $X, Y$ be a pair of its objects. It is said that $\text{Mor}_{\mathcal{C}}(X,Y)$ has again the structure of an $\infty$-category.
What is this structure, how is it defined?
I guess the definition I am familiar with is saying that an $\infty$-category is a simplicial set satisfying a filling property. What are the $n$-simplices of the morphism space? Somehow I thought those should be the $1$-simplices whose face maps give $X$ and $Y$. Is the morphism simplicial set basically the simplicial subset comprised of these simplices?
You are talking about quasicategories as a model for $\infty$-categories. There are a few ways to define the mapping space $\mathrm{Map}_\mathcal{C}(X,Y)$ of a quasicategory $\mathcal{C}$ with given objects $X$ and $Y$ in $\mathcal{C}$ (i.e. $X$ and $Y$ are $0$-simplices of $\mathcal{C}$): the reason is that we only care about the homotopy type of the mapping space (and I use the word ''space'' here actually already to just mean ''homotopy type''), and there can be multiple convenient simplicial sets modelling the same homotopy type. I will of the usual ones the one that is symmetric and gives you the necessary intuition.
So, we can model the mapping space as the following pullback: $$ \require{AMScd} \begin{CD} \mathrm{Map}_\mathcal{C}(X,Y) @>>> \mathcal{C}^{\Delta^1}\\ @VVV @VV{\mathrm{source}\times\mathrm{target}}V\\ \Delta^0 @>{(X,Y)}>> \mathcal{C}\times\mathcal{C} \end{CD} $$ Here, the pullback square is a pullback in the category $\mathsf{sSet}$ of simplicial sets. The object $\mathcal{C}^{\Delta^1}$ is the internal hom of simplicial sets, i.e. $(-)^{\Delta^1}\colon\mathsf{sSet}\to\mathsf{sSet}$ is a right adjoint to the product functor $-\times\Delta^1\colon\mathsf{sSet}\to\mathsf{sSet}$ (I will later say more about how this right adjoint exactly looks like). The internal hom is also functorial in the exponent, so the inclusion $\Delta^0\sqcup\Delta^0\hookrightarrow\Delta^1$ of the endpoints induces a map $\mathcal{C}^{\Delta^1}\to\mathcal{C}^{\Delta^0\sqcup\Delta^0}$. It is abstract nonsense that $\mathcal{C}^{\Delta^0\sqcup\Delta^0}\cong\mathcal{C}\times\mathcal{C}$. The resulting map $$ \mathcal{C}^{\Delta^1}\to\mathcal{C}^{\Delta^0\sqcup\Delta^0}\cong\mathcal{C}\times\mathcal{C} $$ induced by the inclusion $\Delta^0\sqcup\Delta^0\hookrightarrow\Delta^1$ is the map I called $\mathrm{source}\times\mathrm{target}$. This is because $\mathcal{C}^{\Delta^1}$ models for a quasicategory $\mathcal{C}$ the functor $\infty$-category $\mathrm{Fun}(\Delta^1,\mathcal{C})$, where we think of $\Delta^1$ as the one-arrow ($1$-)category. This $\infty$-category is hence just the $\infty$-category of morphisms in $\mathcal{C}$, also called the arrow category. Under this interpretation, the map towards $\mathcal{C}\times\mathcal{C}$ that we defined picks out for a given morphism in $\mathcal{C}$ its source and target. The pullback defining $\mathrm{Map}_\mathcal{C}(X,Y)$ hence informally is the $\infty$-category of morphisms in $\mathcal{C}$ that have source $X$ and target $Y$, and should hence be a good definition of the mapping space in question.
Let us describe the simplices of $\mathrm{Map}_\mathcal{C}(X,Y)$. General nonsense (namely, using the adjunction of the internal hom with the cartesian product) implies that an $n$-simplex of $\mathcal{C}^{\Delta^1}$ is given by some morphism $\Delta^n\times\Delta^1\to\mathcal{C}$ of simplicial sets. We can thus write $(\mathcal{C}^{\Delta^1})_\bullet\cong\mathsf{sSet}(\Delta^\bullet\times\Delta^1,\mathcal{C})$. The $\mathrm{source}\times\mathrm{target}$ morphism corresponds to the restriction morphism $$ \mathsf{sSet}(\Delta^\bullet\times\Delta^1,\mathcal{C})\to\mathsf{sSet}(\Delta^\bullet\times(\Delta^0\sqcup\Delta^0),\mathcal{C})\cong\mathsf{sSet}(\Delta^\bullet\sqcup\Delta^\bullet,\mathcal{C})\cong\mathsf{sSet}(\Delta^\bullet,\mathcal{C})\times\mathsf{sSet}(\Delta^\bullet,\mathcal{C})\cong\mathcal{C}\times\mathcal{C}. $$ Pullbacks in $\mathsf{sSet}$ are taken levelwise (as is the case in any presheaf category), so we find the following explicit description of the model for the mapping space that we defined above: an $n$-simplex in $\mathrm{Map}_\mathcal{C}(X,Y)$ is a morphism $\Delta^n\times\Delta^1\to\mathcal{C}$ such that the restriction $\Delta^n\cong\Delta^n\times\{0\}\to\mathcal{C}$ is the degenerate $n$-simplex on $X$, and such that the restriction $\Delta^n\cong\Delta^n\times\{1\}\to\mathcal{C}$ is the degenerate $n$-simplex on $Y$.
To give an example, we can draw a general $1$-simplex in $\mathrm{Map}_\mathcal{C}(X,Y)$ as follows:
where $f$, $g$ and $h$ are $1$-simplices of $\mathcal{C}$ (i.e morphisms), and $\alpha$ and $\beta$ are $2$-simplices (homotopies), and the left-hand and right-hand identity signs stand for the degenerate $1$-simplices on $X$ and $Y$ respectively.
As I said earlier, there are more simplicial sets that model the same homotopy type, and sometimes are easier to use for technical reasons. However, when using $\infty$-categories, you would want to move as soon as possible to statements that do not depend on the models you use for these mapping spaces, as in actual $\infty$-category theory everything is only defined up to homotopy anyway.
As for the question why $\mathrm{Map}_\mathcal{C}(X,Y)$ is an $\infty$-category: this is easiest using the theory of inner fibrations. A simplicial set $S$ is an $\infty$-category iff the morphism $S\to\Delta^0$ is an inner fibration. General theory of quasicategories gives us that the map $\mathrm{source}\times\mathrm{target}\colon\mathcal{C}^{\Delta^1}\to\mathcal{C}\times\mathcal{C}$ is an inner fibration. Moreover, inner fibrations are stable under pullback, and this would then show that our definition of $\mathrm{Map}_\mathcal{C}(X,Y)$ gives an $\infty$-category. It is even an $\infty$-groupoid/Kan complex, i.e. an actual space. This is easiest to see using the concept of conservative morphsims. General theory of quasicategory shows that the map $\mathrm{source}\times\mathrm{target}\colon\mathcal{C}^{\Delta^1}\to\mathcal{C}\times\mathcal{C}$ is conservative (which is to be seen as an instance of the statement that isomorphisms in functor categories are exactly the morphisms that are levelwise isomorphisms). Conservative morphisms are also stable under pullback, and this shows that all morphisms in the $\infty$-category $\mathrm{Map}_\mathcal{C}(X,Y)$ are isomorphisms.