Please help me to identify what the asterisk symbolizes in the following:
I would have thought it would mean a dual space, but the author uses $'$ to symbolize the dual space. In fact, the preceding paragraph is this: 
If it is the dual space, I do not understand how $l_q$ is the dual of $l_p$. The asterisk is not introduced earlier in the notes.
The algebraic dual space of a vector space $V$ is the space of the linear functionals on $V$. If $V$ is finite dimensional than all linear functional are continuous but this is not true, in general, if $V$ is infinite dimensional. So, in this case we introduce the notion of continuous (or topological) dual space, that is the subspace of the algebraic dual that contains all the continuous functional. It seems that in your case (note that the $l^p$ spaces are infinite dimensional) the algebraic dual is indicated by $V'$ and the topological dual by $V^*$.
The proof that the (topological) dual of $l^p$ is $l^q$ such that $1/p+1/q=1$ is classical and you can find it on any book on functional analysis. You can also see a sketch here.