Let $X_n=\sum\limits_{k=1}^n Y_k = X_{n-1} + Y_n$ and $X_0=0$, $(Y_n)$ i.i.d with
$P(Y_n=1)=p=1-P(Y_n=-1)=1-q, p \in (0,1)$, $(X_n)$ is a random walk on $\mathbb{Z}$.
Why is this Markov chain irreducible? How do we know that all states (integers) communicate with each other? How do we show this intuitively?
Two states $x,y$ communicate if it is possible to get from $x$ to $y$ with positive probability and viceversa. If all the states communicate then the chain is said to be irreducible.
The chain you described is irreducible because there is always a non-zero probability to go from every state $x \in \mathbb{Z}$ to another $y \in \mathbb{Z}$ in a finite number of steps. In particular, from $x$ you can get to $y$ in $y-x$ steps with probability $p^{(y-x)}$ if $x \leq y$ or with probability $(1-p)^{(y-x)}$ if $x > y$.
You can think of a random walk as a process that takes steps by tossing a biased coin: a Head corresponds to moving up, a Tail corresponds to moving down. There is always a non-zero probability of obtaining enough Heads (or Tails) in a row such that you can reach any target position from any initial position.
