The two expressions denote a change of variable (change of scale): they apply
$$
\eqalign{
& 1 = \int_{\,r\, = \,0}^{\;\infty } {P(r)dr} = \int_{\,r\, = \,0}^{\;\infty } {LP(r)d\left( {{r \over L}} \right)} = \cr
& = \int_{\,r\, = \,0}^{\;\infty } {LP\left( {L{r \over L}} \right)d\left( {{r \over L}} \right)}
= \int_{\,\tilde r\, = \,0}^{\;\infty } {LP\left( {L\tilde r} \right)d\tilde r} = \cr
& = \int_{\,\tilde r\, = \,0}^{\;\infty } {\tilde P\left( {\tilde r} \right)d\tilde r} \cr}
$$
to pass from $r,P(r)$ to $\tilde r,\tilde P\left( {\tilde r} \right)$ by putting
$$
\left\{ \matrix{
\tilde r = r/L \hfill \cr
\tilde P\left( {\tilde r} \right) = LP\left( {L\tilde r} \right) = LP\left( r \right) \hfill \cr} \right.
$$
This is a totally licit and very common operation done in probability, for example when reconducing
a Normal distribution with a given $\sigma$ to the standard one.
They explain that such a "standardization" allows to simplify (in some cases) the expressions by "absorbing"
the $L$ parameter, which is in fact a scale parameter. The parallel with the Normal helps
to understand why.
That premised, concerning your doubt on the average,
$$
\int_{\,\tilde r\, = \,0}^{\;\infty } {\tilde r\,\tilde P\left( {\tilde r} \right)d\tilde r}
$$
gives of course the average of $\tilde r$ , denoted as $ \left\langle {\tilde r} \right\rangle$
which tied to $ \left\langle {r} \right\rangle$ by
$$
\left\langle {\tilde r} \right\rangle = \left\langle {r/L} \right\rangle = \left\langle r \right\rangle /L
$$
In fact, soon after eq. (31) they speak of avg.$\tilde r$ as the "average relative number of consecutive cookies ..":
relative is understood to refer to $/L$, and actually immediately below they give
$\left\langle {\tilde r} \right\rangle = \left\langle {r/L} \right\rangle = \cdots $.
Addendum
Going back to eq.(30) reported at the beginning of your post
$$
P(r) = 2q{{\Gamma (L)} \over {\Gamma (L - 2q)}}{{\Gamma (L + r - 1 - 2q)} \over {\Gamma (L + r)}}
$$
The average number of $r$ would be given by
$$
\left\langle r \right\rangle = \sum\limits_{0\, < \,r} {r\,P(r)}
= 2q{{\Gamma (L)} \over {\Gamma (L - 2q)}}\sum\limits_{0\, \le \,r} {\left( {r + 1} \right)\,{{\Gamma (L + r - 2q)} \over {\Gamma (L + r + 1)}}}
$$
In the above $q$ is a real number in the range $(0,1)$; the sum above
can be expressed by means of the Gaussian Hypergeometric Function as
$$
\left\langle r \right\rangle = {{2q} \over L}\;{}_2F_{\,1} \left( {2,\,L - 2q\,;\;L + 1\,;1} \right)
$$
which, in virtue of the Gaussian theorem gives simply
$$
\eqalign{
& \left\langle r \right\rangle = {{2q} \over L}{{\Gamma (L + 1)\Gamma ( - 1 + 2q)} \over {\Gamma (L - 1)\Gamma (1 + 2q)}}
\quad \left| {\,0 < {\mathop{\rm Re}\nolimits} \left( { - 1 + 2q} \right)} \right.\quad = \cr
& = \left\{ {\matrix{ {{{\left( {L - 1} \right)} \over {\left( {2q - 1} \right)}}}
& {\left| \matrix{ \;1 \le L \hfill \cr \;1/2 < q \hfill \cr} \right.} \cr \infty
& {\left| \matrix{\;1 \le L \hfill \cr \;q \le 1/2 \hfill \cr} \right.} \cr } } \right. \cr
}
$$
which, for large $L$, correspond to eq.(32).
To this regard we shall note that:
- the summand $\left( {r + 1} \right)\,{{\Gamma (L + r - 2q)} \over {\Gamma (L + r + 1)}}$ has a series expansion at $r=\infty$ which
is $1/r^{2q} + O(1/r^{2q+1})$ and the sum is therefore convergent for $1<2q$;
- the Hypergeometric $ {}_2F_{\,1} \left( {a,\,b\,;\;c\,;z} \right)$ has a singularity at $z=1$, so that
its value there shall be taken in the limit with due restrictions;
- the restrictions are those provided for the validity of its conversion into the fraction with Gammas, i.e. $0<1-2q$.
