In the proof of Chapman Kolmogorov Equation

$p_{ij}^{(m+n)}=\sum_{k=0}^{\infty}p_{ik}^{(n)}p_{kj}^{(m)}$

Proof:

$p_{ij}^{(m+n)}=P[X_{m+n}=j|X_0=i]$
By the total probability it says
$P[X_{m+n}=j|X_0=i]=\sum_{k=0}^{\infty} P[{X_{m+n}=j,X_n=k |X_0=i}]$

I don't understand how $\sum_{k=0}^{\infty} P[{X_{m+n}=j,X_n=k |X_0=i}]$ was obtained.What is $X_n=k$?

In the proof of Chapman Kolmogorov Equation

$p_{ij}^{(m+n)}=\sum_{k=0}^{\infty}p_{ik}^{(n)}p_{kj}^{(m)}$

Proof:

$p_{ij}^{(m+n)}=P[X_{m+n}=j|X_0=i]$
By the total probability it says
$P[X_{m+n}=j|X_0=i]=\sum_{k=0}^{\infty} P[{X_{m+n}=j,X_n=k |X_0=i}]$.

I don't understand how $\sum_{k=0}^{\infty} P[{X_{m+n}=j,X_n=k |X_0=i}]$ was obtained. What is $X_n=k$?