So a $t$-star is the graph you get by taking one central node and connecting $t$ neighbors to it. Equivalently, a $t$-star is the complete bipartite graph $K_{1,t}$ if you wish to express it like that as your definition does.

A *$t$-star packing* then is a bunch of such $t$-stars found from some graph $G$ so that their nodes don’t overlap. As a simple example, consider the 4-cycle. You can find a 1-star packing of size 2 in it: take one edge as the other 1-star, and the other edge as the other 1-star.

The packing described above is *maximal*: you can’t extend it by adding a new member (i.e., a third 1-star) into it. It also happens to be *maximum*, i.e., it is the largest such packing you can find. I encourage you to think about what the difference between a *maximal* and *maximum* packing is, i.e., find an example where they differ.

I don’t what *proper* here means, but it is likely defined in your source.