What does it mean to observe a Markov Chain after a certain kind of transition?

If the state space of the Markov chain is $S$ and you observe the chain only when it is in $U\subseteq S$, then the result is still a Markov chain but with different transition probabilities.

Let $(X_n)_{n\geqslant0}$ denote the original Markov chain and $T=\inf\{n\geqslant1\mid X_n\in U\}$. Then the transition probabilities of the new Markov chain are such that, for every $x$ and $y$ in $U$, $$ Q(x,y)=P(X_T=y\mid X_0=x). $$ In general, the new transition probabilities $Q(x,y)$ are a complicated functional of the transition probabilities of $(X_n)_{n\geqslant0}$ and of the disposition of $U$ in $S$.

Here is an example. Assume $(X_n)_{n\geqslant0}$ is the symmetric $\pm1$ random walk on $\mathbb Z$, with transition probabilities $P_x(X_1=x+1)=P_x(X_1=x-1)=\frac12$ for every $x$ in $\mathbb Z$. Let $U\subseteq\mathbb Z$ with $U=\{x_k\mid k\in\mathbb Z\}$ and $x_k<x_{k+1}$ for every $k$. Then, starting from $x_k$, the new Markov chain can only jump to a vertex $x_j$ such that $|j-k|\leqslant1$, and the transition probabilities are $$ Q(x_k,x_{k-1})=\frac1{2(x_k-x_{k-1})},\quad Q(x_k,x_{k+1})=\frac1{2(x_{k+1}-x_{k})}, $$ and $$ Q(x_k,x_{k})=1-\frac1{2(x_k-x_{k-1})}-\frac1{2(x_{k+1}-x_{k})}. $$ Hence the new chain behaves like a random walk on $\mathbb Z$ with jumps from $k$ to $k$ or $k+1$ or $k-1$, and basically any kind of transition probabilities.

As Didier points out, the transition probabilities for the new chain are complicated. But for a finite state space Markov chain, we can let linear algebra do the work. First partition the transition matrix $P$ as shown:

$$P= \matrix{&\hskip-15pt \matrix{&\small U&\small U^c}\cr\matrix{\small U\cr\small U^c}&\hskip-10pt\pmatrix{A&B\cr C&D}}.$$

That is, the matrix $A$ has the transitions from $U$ to $U$, the matrix $B$ has the transitions from $U$ to $U^c$, etc. The transition matrix for the new chain with state space $U$ is $$P^U=A+B(I-D)^{-1}C.$$

The inverse of $I-D$ will exist if it is possible to reach $U$ from any state in $U^c$.