This kind of trick is common when thinking about tensors.
Let $v$ be a vector, $\alpha$ a one-form, and $c$ a number. A vector gives a map from one-forms to numbers,
$$\alpha \mapsto v(\alpha)$$
but it also gives a map from numbers to vectors,
$$c \mapsto c v.$$
Similarly, a $(1, 1)$ tensor $T$ gives a map from a vector and covector to numbers,
$$(\alpha, v) \mapsto T(\alpha, v).$$
But it also gives a map from vectors to vectors,
$$v \mapsto T(\cdot, v).$$
To see that the output here is a vector, note that it gives a map from one-forms to numbers,
$$\alpha \mapsto T(\alpha, v).$$
By continuing with this, you can also use a $(1, 1)$ tensor to define a map from covectors to covectors, or numbers to a vector and covector, and so on.
To avoid having to associate the exact same object with a ton of different maps, physicists use index notation. In this notation, the five preceding maps take the form
$$\alpha_i \mapsto v^i \alpha_i, \quad c \mapsto c v^i, \quad (\alpha_i, v^j) \mapsto T^i_j \alpha_i v^j, \quad v^i \mapsto T^i_j v^j, \quad \alpha_i \mapsto (T^i_j v^j) \alpha_i.$$
From this you can get the general pattern, which is that a $(n, m)$ tensor can act on a $(n', m')$ tensor to yield a $(n+n'-k, m+m'-k)$ tensor where $k$ is the number of upstairs/downstairs index pairs that are contracted.