Some preamble first, to set the scene. For this emitter follower, \$V_{OUT} = V_{IN} - 0.7V\$:
simulate this circuit – Schematic created using CircuitLab
We'll assume that V1 (the source of input \$V_{IN}\$) never exceeds V2 (the source of supply voltage \$V_{CC}\$, because if it did, then the base-collector junction of transistor Q1 becomes forward biased, and the operation of this circuit moves into a completely different regime, in which nothing I'm about to say is valid any more.
We'll also assume that V1 never falls below +0.7V, because if that happens the base-emitter junction is no longer forward biased, and Q1 is in cut-off.
Another assumption I'll make is that Q1's current gain \$\beta\$ is very high (say, 100 or so) meaning that collector current \$I_C\$ and emitter current \$I_E\$ are roughly equal. Base current \$I_B\$ is so small that its exact value is unimportant for this argument.
So the conditions are:
$$ +0.7V < V_{IN} < V_{CC} $$
$$ I_E \approx I_C $$
Under these assumptions, the base-emitter junction is always forward biased, passing a tiny base current \$I_B\$ so \$V_{BE}\approx 0.7V\$. We can apply KVL to obtain an expression for output \$V_{OUT}\$ in terms of \$V_{IN}\$:
$$ V_{OUT} = V_{IN} - 0.7V $$
This already suggests that \$V_{OUT}\$ is independent of \$V_{CC}\$, and this is backed up when we write \$V_{OUT}\$ in terms of emitter current \$I_E\$ through \$R_1\$, using Ohm's law:
$$
\begin{aligned}
V_{OUT} &= 0V + V_{R1} \\ \\
&= 0V + I_ER_1 \\ \\
&= I_ER_1
\end{aligned}
$$
Still no mention of \$V_{CC}\$.
Since \$V_{BE} \approx 0.7V\$, one might expect the transistor to be saturated, forcing \$V_{OUT}\approx V_{CC}\$. However that's not the case, so let's see why.
Imagine that for some reason \$V_{OUT}\$ decreases. In other words, the voltage \$V_{R1}\$ across R1 decreases. If \$V_{IN}\$ stays fixed at +5.0V, this would increase \$V_{BE}\$. Remembering that the base-emitter junction is just a diode, this would in turn increase current \$I_B\$ through it. Collector current \$I_C\$ would also increase, because \$I_C=\beta I_B\$, and so will \$I_E\$.
But if \$I_E\$ increases, the voltage across R1 must also increase, in opposition to the change that caused \$V_{R1}\$ to fall in the first place.
A similar thing happens if \$V_{OUT}\$ were to rise somehow. By the same reasoning, this would result in a decrease of \$V_{BE}\$, accompanied by a corresponding decrease in \$I_B\$ and \$I_E\$, finally causing \$V_{R1}\$ (and of course \$V_{OUT}\$) to fall, in opposition to the change that caused \$V_{R1}\$ to rise in the first place.
A pattern is emerging, that this arrangement strongly opposes any attempt to change \$V_{OUT}\$. Whatever we do to change \$V_{OUT}\$ results in a response by the transistor to oppose that change, a kind of "negative feedback" inherent in any emitter-follower configuration. This is its chief purpose.
One other thing also emerges from all this, that the transistor is never fully switched on or off. \$V_{BE}\$ may be close to 0.7V, but it's never quite large enough to fully switch on the transistor (saturation), but also never small enough to switch it complete off (cut-off). The system is always in a state of equilibrium in which \$V_{BE}\$ is exactly the right value to produce \$V_{OUT}=V_{IN}-0.7V\$. If \$V_{BE}\$ is perturbed one way or the other, the consequent change in collector and emitter currents immediately undo that perturbation. Inherent negative feedback.
This all relies on the absence of a sharp switching threshold at \$V_{BE}=0.7V\$. If the transistor were to switch between cut-off and saturation with no in-between state, then this behaviour wouldn't be possible. The transistor is required to go between cut-off and saturation over a range of \$V_{BE}\$, say 0.65V to 0.75V for this to work. A simple experiment shows this in action:
simulate this circuit
In this circuit I am not applying applying a base potential with respect to ground (0V), I am applying a potential difference directly between base and emitter, explicitly controlling \$V_{BE}\$. This permits me to see how collector current varies as a function of \$V_{BE}\$ directly:
As you can see, there is a range of values of \$V_{BE}\$, 0.6V to 0.75V, over which collector current goes from minimum 0A to maximum \$\frac{12V}{R_1}=12mA\$, and the emitter follower always finds an equilibrium somewhere on the curve in between those extreme cut-off and saturation states.
In other words, it is always operating in its active region. Consequently \$V_{OUT}\$ can be (and always is) less than \$V_{CC}\$.