Why is the set $ \mathbb{Z}_{+} \times \{a, b \}$ limit point compact?

Say $Y=\{a,b\}$. If $S$ is a subset of $\Bbb Z_+\times Y$ and $(n,a)\in S$, then $(n,b)$ is a limit point of $S.$ Conversely, if $(n,b)\in S$, then $(n,a)$ is a limit point of $S.$

There is also the notion of countable compactness: A space is called countably compact if every countable open cover has finite subcover.

Every countably compact space is limit point compact. The converse can fail, as your example demonstrates. This is mainly due to the absence of the $T_1$ property, as every limit point compact $T_1$ space is also countably compact.
Another example of such a space is $X=\Bbb N$ with the topology generated by the sets $A_n=\{k\mid k< n\}$ for all natural $n$. It is limit point compact but not countably compact.