Presumably, this is a sequence indexed by some ordinal number. To be precise, an ordinal number $\kappa$ (which in the standard construction is identified with the set of all ordinals less than $\kappa$) can be given the order topology, and a transfinite sequence indexed by $\kappa$ in a space $X$ is just a continuous map $\kappa \to X$.
A sequence indexed by a finite ordinal $n$ is just an $n$-tuple of points. Ordinary sequences are indexed by the ordinal $\omega$. A sequence indexed by $\omega + 1$ is an ordinary sequence with a limit point. And so on.
Though I've never seen this definition before, here's what I'd guess is the reasoning behind it. We're used to talking about topological conditions like closure, convergence, and so on in terms of sequences, and in metric spaces or more generally first countable spaces, this is all we need. In 'bigger' spaces, the topology of the space isn't really captured by mere countable sets of points, so we need to replace the familiar notion of 'sequence' with something more general. This turns out to be a thing called a net, which is a sequence indexed by a directed set. So while it's not generally true that every limit point of a set in a topological space is a limit of a sequence in that set, it is true that it's the limit of a net in that set.
Now, directed sets are only partially ordered, so dealing with nets might end up being too general. In a radial space, you only need to deal with nets indexed by well-ordered sets -- that is, with transfinite sequences indexed by ordinals.