Idea: we construct , on each interval $[n,n+1]$ , a countably-infinite collection of linear functions (linear functions $ax+b$ , with both a,b in $\mathbb Q$) mapping $\mathbb Q$ to itself, that extend to continuous functions $f: \mathbb R \rightarrow \mathbb R$. Linear functions extend because they are uniformly-continuous on a dense subset, and uniform continuity is sufficient to extend a function from a dense subset into the whole space. Then we extend each of these functions from $[n,n+1]$ to $[n+1,n+2]$ continuously. Then, each of the $|\mathbb Q|$ functions on $[n,n+1]$ can be (linearly)extended to $[n+1,n+2]$ in $|\mathbb Q|$ ways, so that the total cardinality is $|\mathbb Q| \times |\mathbb Q| \times....$ $|\mathbb Q|$ times.
Consider the integers $ \mathbb Z$ . We construct an uncountable collection of linear maps from $\mathbb Q $ to $\mathbb Q$, and we use the fact that linear maps, being uniformly-continuous on the dense subset $\mathbb Q$ of $\mathbb R$, extend to a continuous map $f: \mathbb R \rightarrow \mathbb R$ .Start at, say $0$. Then, following the idea of the link, any line thru the point $0$ with rational slope maps $\mathbb Q$ to $\mathbb Q$: take $px+q$ , with $p,q$ both in $\mathbb Q$, since Rationals are closed under multiplication, then $px$ is Rational as a product of Rationals, and when we add $b$ to it we have a sum of Rationals, which is Rational. Notice this choice of line can be made in $|\mathbb Q|= \aleph_0$ ways . Now, extend the function at $x=1$ , starting at the image $a(1)+b$ , and then extend the same way from $x=2 $ to $x=3$ , i.e., you defined $a'x+b'$ in $[1,2]$ to be $a'x+b'$ , with both $a',b'$ Rational. This means that each of the $|\mathbb Q|$ choices in each of the interval $[n,n+1]$ can be combined with $\mathbb Q$ choices in $[n+1, n+2]$ , for all integers $n$. So you have a total of $|\mathbb Q|\times |\mathbb Q |\times...|.....$ , all of this $|\mathbb Q|$ times, which gives you an uncountable collection of functions $f: \mathbb Q \rightarrow \mathbb Q$ , that extend to continuous functions $F: \mathbb R \rightarrow \mathbb R$, and you're done.