In general, once you have two totally ordered set $A$ and $B$ you can form another totally ordered set $A+B$ by declaring that for every $a\in A$ and $b\in B$ $a<b$
In particular $\omega+1 =\{0<1<2<3<\cdots<\omega\}$
I don't know what $2^{<\omega}$ is, but I think that the order-preserving bijection $\omega\to \omega^*$ that JDH is describing is clear enough
The point is that in $(\omega+1)*$ with the subword ordering every object has finitely many objects below, while this is not true in $\omega +1$
Any order preserving bijection $f:\omega+1\to (\omega+1)^*$ has to send the maximum of $\omega+1$ to some element with infinitely many predecessors, but such an element does not exists
That is, if $f:\omega+1\to (\omega+1)^*$ is an order preserving bijection, then every $f(n)$ is a distinct subword of $f(\omega)$, but this is impossible since $f(\omega)$ does not have infinitely many subwords
In all seriousness, you should try and give that question some time: read more about the subject and perhaps you will be able to get answers on your own.
@EspenNielsen I'm rather a fan of the book "$H_\infty$ Ring Spectra and their Applications" (also by May and collaborators). Chapter VIII section 3 there (by McClure) specifically treats the homology Dyer-Lashof operations
Section IX.1 in the same book deals with homology operations over more general spectra
