Let the set $K_i = (S, I_i)$ be a matroid for each $i \in \{1 \ldots k\}$.
We define $K = (S, I) $ where $I = \{ X \subset S $ | $ X \in \bigcap_{i=1}^k I_i\}$
The claim now is that $K$ is a k-matroid. Which means that $K$ is an independent set and for all $X, Y \subset S$ with $|X|>k|Y|$ there exists an $s \in X\setminus Y$ such that $Y \cup \{s\} \in I$.
I don't understand why the last property is true.