What are the cycles in this graph, and what are their sizes? I have the following graph $G$. I'd like to find how many cycles there are and what their sizes are.

Please correct me if I am wrong: in this graph there are $2$ cycles, $\text{1-2-3-4-5-1}$ with size $5$, and the other $\text{4-5-4}$ with size $2$.
Is that true? As mentioned in a theorem:

If $G$ is simple and deg$^+(v)\geq k\geq 1$, then there is a simple cycle of at least size $k+1$ ...
  where $deg^+(v)$  is  the out-degree of v.
  I did not find $(k+1)$ here?

how can I choose value of k in the second Graph:


The hypothesis of the theorem should be that $\deg^+(v)\ge k\ge 1$ for all nodes $v$ of the graph. In your graph nodes $1,2,3$, and $4$ have out-degree $1$, and node $5$ has out-degree $2$, so we must choose $k=1$. The theorem then guarantees that there is a cycle of length at least $2$, which is true: as you point out, there are two of them, one of length exactly $2$ and one of length $5$.
Note that the theorem does not guarantee a cycle of length exactly $k+1$, though in this case there is one; it just says that there is a cycle whose length is $\ge k+1$.
