How to calculate growth function for a threshold function
Yes, your interpretation is correct. 

Each member of $H$ is one such function, they are parametrized by $i$, the feature selected and $\theta$, the chosen threshold. Different $\theta$ might corresponds to the same function but effectively there are only $m+1$ such function for each $i$. Hence, the set $H$ consists of at most $(m+1)N$ elements.