Тhe answers so far are perfect but specific; they explain how the circuit is made. For the true understanding, however, this is not enough. It is also necessary to explain why (for what purpose) this was done in this way and to consider its applications in practice. I will try to satisfy this need with my answer below.
I was pleasantly surprised to see that someone, willingly or unwillingly, came up with the same idea as me many years ago when I used it to make a battery backup power supply. It is based on an inherent BJT property sometimes causing strange problems - when the base-emitter junction is fully on, it actually connects, like a forward biased diode, the base to the emitter... and the input voltage source is directly connected to the emitter. This situation can happen in an emitter follower (such as here), if the emitter ceases, for some reason, to follow the base. Usually, this is an undesired situation but there are also its useful applications like in the link above.
So, the OP's question is about an emitter follower considered in two situations - without supply voltage ("damaged") and with supply voltage ("restored"). Considering them one by one, we can explain the meaning of this circuit when used as a buffer.
I will use two figures from the link above to illustrate my explanations. There the invisible electrical quantities are visualized as follows. Voltages are represented by vertical segments (voltage bars) with proportional height in red. They are summed (subtracted) geometrically, according to KVL (this clearly shows the relationship between voltages). Thus the set of voltage bars on the circuit diagram can be considered as a snapshot of the voltage relief. Current paths are shown by closed lines (current loops) starting from the positive terminal of the power supply and ending at its negative terminal. Current magnitudes are hinted by the line thickness.
"Damaged" emitter follower
If there is no supply voltage (the collector is unconnected), there is no emitter follower since the transistor is not a transistor but rather a diode - Fig. 1. Only its base-emitter junction is used as such a "diode"... and the input voltage VB is transferred directly through it to the load in the emitter (the 1 k resistor in the OP's picture).
Fig. 1. An emitter follower without supply voltage
Since the battery supplies the load through the base-emitter junction the whole load current IL flows through the junction. As though the input voltage source is directly connected to the load (only a small voltage drop VBE is subtracted from the input voltage). Usually, this is not a desired situation since the load can be too heavy for the input voltage source and can overload it.
"Restored" emitter follower
So let's restore the supply voltage (connect the collector) - Fig. 2.
Fig. 2. An emitter follower with supply voltage
The transistor "becomes a transistor" and begins acting as a device with negative feedback... or as a "being" implementing some goal. "Seeing" the significant base-emitter voltage, it begins increasing its collector current (IL) so the voltage drop across the load (VL = IL.RL) increases. Since the base-emitter voltage is a difference between the constant input voltage VB and the increasing output voltage VL, it begins decreasing (VBE = VB - VL)... and finally the equilibrium is reached by the negative feedback. VL will be (slightly) higher than in the case of the "damaged" follower; so the load current (IL = VL/RL) will be slightly higher as well.
Now let's look at the situation from another perspective. The transistor passes its collector current through the load in parallel to the base current thus "helping" it. The input battery provides only the small base current needed to set the beta times higher collector current and, accordingly, the load current needed. As though, the battery tries to supply the load by injecting the small base current through the base-emitter junction to the load... and the power supply, by the help of the transistor, "helps" it by adding beta times bigger collector current. Figuratively speaking, you can think of the collector-emitter part as a "big brother" that helps the "little brother" (base-emitter junction:) This metaphor can help to intuitively understand the main idea behind the emitter follower used as a buffer.