I'm also starting to read this book. In this definition 1.13, it is written:
The operations of addition, scalar multiplication, and multiplication of functions induce on $\tilde{F}_{m}$ the structure of an algebra over $\mathbb{R}$.
Then you need to know the definition of $\mathbb{R}$-algebra, i.e., a structure of an algebra over $\mathbb{R}$. You'll find a good definition in Algebra, by Birkhoff-Mac Lane. OK, let's continue: since $\tilde{F}_{m}$ is $\mathbb{R}$-algebra, it is in particular a ring. Therefore $F_{m}$ is an ideal of $\tilde{F}_{m}$ with respect to the ring structure that exists on $\tilde{F}_{m}$. The same applies to $F_{m}^{k} \subset \tilde{F}_{m}$; it is also an ideal, but a little more complicated:
$F_{m}^{k}=\left\lbrace [h]=a_{1}[g_{1}]+\cdots+a_{n}[g_{n}]\;\middle\vert\; {{n\in \mathbb{N}, a_{i}\in \mathbb{R},\text{ and }[g_{i}]=[f_{1}][f_{2}]\cdots[f_{k}],}\atop {\text{where }[f_{j}]\in F_{m}, 1\leq j\leq k,\text{ and }1\leq i\leq n }}\right\rbrace.$ I hope I have helped; be sure to read the definition of $\mathbb{R}$-algebra, there are two different structures within the same set.