The article is indeed confusing. Let me focus on the $p=1/2$ case. Note first that the tent map is not invertible, as shown in the picture below
from wikipedia).

The tent map has two inverse branches. What is (almost) invertible is the conjugacy between the shift map on the symbolic space and the tent map. This conjugacy can be defined using the binary expansion of reals in the interval $[0,1]$ (for $p=1/2$). This is slightly easier to define for the doubling map $x \mapsto 2x \ mod \ 1$ and is done as follows.

$$ \matrix{\varphi : &\{0,1\}^{\bf N} & \rightarrow & [0,1]\cr
& (a_i)_{i\in {\bf N}} & \mapsto & \sum_{i=0}^{\infty} {a_i\over 2^{i+1}} \cr}$$

This is best represented in the diagram below.

$$
\matrix{
\{0,1\}^{\bf N} &
\xrightarrow{shift}
&
\{0,1\}^{\bf N} \cr
\ {\llap{\varphi}\left\downarrow
\right.} &
&
\ {\left\downarrow
\right.\rlap{\varphi}} \cr
[0,1] &
\xrightarrow{doubling} &
[0,1] \cr}
$$

Note that there is also a semiconjugacy from the doubling map to the tent map from the relation $tent \circ doubling = tent \circ tent$.

$$
\matrix{
[0,1] &
\xrightarrow{doubling}
&
[0,1] \cr
\ {\llap{tent}\left\downarrow
\right.} &
&
\ {\left\downarrow
\right.\rlap{tent}} \cr
[0,1] &
\xrightarrow{tent} &
[0,1] \cr}
$$
It's better to define the symbolic model for the tent map directly though so that the semi-conjugacy is invertible outside a countable set of points.

See the classical book of Hirsh, Smale and Devaney, "*differential equations, dynamical systems and an introduction to chaos*" chapter 15 for a nice discussion of the tent map and its symbolic model in the case $p=1/2$. You will find there a far better explanation of the model. Note that getting the symbolic sequence associated to a point just amounts to iterate the point and checking if the iterate is on the left interval $[0,1/2]$ (-> bit 0) or on the right interval $[1/2,1]$ (-> bit 1).

My thanks to Anthony Quas for correcting me about the conjugacy.