Help with explanation on this transistor behaviour
RC timing

Ignoring the BJT and LED portions of the circuit, the following shows you how to approximate the RC charging process. The left side is from your schematic. The right side is the Thevenin equivalent (same thing, just slightly simplified):







simulate this circuit &ndash; Schematic created using CircuitLab

Looking at the right side, find that \$V_\text{TH}=9\:\text{V}\cdot\frac{R_2}{R_1+R_2}\$ and that \$R_\text{TH}=\frac{R_1\cdot R_2}{R_1+R_2}\$. It's common practice to invent (create) a new variable: \$\tau=R_\text{TH}\cdot C_1\$ (multiplying resistance in Ohms times capacitance in Farads yields units in seconds.) This is done because much is known from this simplification. In \$1\tau\$ the voltage will rise to about 63% of the supply voltage (the Thevenin value in this case); in \$2\tau\$ it's 86%; and in \$3\tau\$ it's about 95%. Most of the work has been done in about \$3\tau\$ and this fact can be used as a guide to determine when there's very little change remaining.

(This ignores the fact that the base of the BJT will also require some current and therefore affects the timing just mentioned. But if \$R_\text{TH}\$ is low enough in value, the base current of the BJT won't matter enough to worry about.)

If the voltage across \$C_1\$ starts at \$0\:\text{V}\$ when \$t=0\$, then \$V_{\text{base}\left(t\right)}=V_\text{TH}\cdot\left(1-e^{\left[\frac{-t}{\tau}\right]}\right)\$.

(This allows you to work out what values of \$V_\text{TH}\$, \$R_\text{TH}\$ and \$C_1\$ you want, once you have decided how much current your LEDs will require and what kind of base current will be needed for that.)



Adding the emitter-follower BJT

Let's add the so-called emitter-follower BJT arrangement, now:







simulate this circuit

Here, the BJT's emitter will simply "follow" the base voltage, less one diode drop. So as the voltage at the base follows the \$V_{\text{base}\left(t\right)}\$ equation and rises upward, so also will the emitter rise upward too. This arrangement, though, uses the collector to source a lot of added current, if needed, because the collector can source up to \$\beta\$ times the base current. So a tiny base current permits a very large, added collector current. This means that the emitter can source a great deal of current if the load attached to the emitter wants it, without significantly affecting the emitter voltage which is following the base voltage. At this point, you can consider adding some kind of load to the emitter, knowing that the BJT will boost the current compliance.

The only caveat here is that the emitter follows the base and cannot exceed a voltage that is higher than one diode drop below the base voltage. And since the base voltage cannot exceed \$V_\text{TH}\$, the emitter voltage cannot exceed \$V_\text{TH}-V_\text{BE}\$. Luckily, you can arrange for different values of \$V_\text{TH}\$ by modifying the relationship between \$R_1\$ and \$R_2\$. But this is still a limitation to be aware of.

Emitter-follower BJT with added load

Here is the effective circuit you'd started with:







simulate this circuit

So what happens now?

Well, the LED diode drops must be met before anything happens. For old-style, typical red LEDs operating at \$20\:\text{mA}\$, this is about \$2\:\text{V}\$ per LED. For high intensity modern LEDs, this can easily reach \$3.8\:\text{V}\$ per LED, or more (when the currents are rather high, too.) So the base voltage must exceed the LED requirements (you show two of them, so twice the voltage requirement of just one.) It must also exceed the \$V_\text{BE}\$ required by the BJT, itself, as well. Any residual voltage that is left, after accounting for the two LEDs and  \$V_\text{BE}\$, will appear across \$R_3\$ and this will set the current.

If you think about this for a moment, you'll realize that it will take some time for the base voltage to reach a point where the sum of the two LED voltages and  \$V_\text{BE}\$ is finally met. Up until this point, nothing much happens and the LEDs will not be lit up. Once it is reached, \$R_3\$ will begin to limit the current as the base voltage continues upward (assuming it does so.) So the LED currents will "follow" this rise, once the voltage requirements have been met.

Ultimately, the final current in the LEDs will be \$I_\text{LED}=\frac{V_\text{TH}-2\cdot V_\text{LED}-V_\text{BE}}{R_3}\$.

A significant problem in this design is that the final current depends a great deal on (mostly) \$V_\text{LED}\$ and the BJT's \$V_\text{BE}\$ (less so.) And since the value of \$V_\text{LED}\$ can vary quite a bit from one LED to another (at the same current), this is a source of management error in the circuit. It's not terrible, since most folks won't really notice too much. But if \$V_\text{TH}-2\cdot V_\text{LED}-V_\text{BE}\$ doesn't leave much margin for \$R_3\$ to do its work, then the results can vary quite a lot from one circuit's part selections to another circuit with different parts.

This means this isn't really a good circuit. It may works after a fashion, assuming \$V_\text{TH}\ge 2\cdot V_\text{LED}+V_\text{BE}\$. But you may not get consistent results, time after time. Also, we've completely neglected what happens when the power supply is turned off (how does \$C_1\$ discharge itself so that it starts again where it left off, the last time?)



Simple improvement

One way to improve the circuit is to remove the dependency of the LED current upon the LED voltage. To do that, simply move the two LEDs into the collector of the BJT:







simulate this circuit

This only works well if your BJT collector can remain at a voltage that is above \$V_\text{TH}\$. But since you can design things so that \$V_\text{TH}\lt 9\:\text{V}-2\cdot V_\text{LED}\$, it doesn't pose an insurmountable problem.

Here, the only delay until the LEDs start experiencing current is the short delay required until the voltage on \$C_1\$ (the base of \$Q_1\$) exceeds \$V_\text{BE}\$. (This is a shorter delay than before.)

Also, this removes a dependence upon the variability of your LED voltages. So that's a good thing for the circuit. So this is actually an improvement over the prior design.

Final commentary

There are many additional ideas you could apply, if you wanted to make the circuit still better.

One problem is how to deal with what happens when the power supply is turned off. So far, we've assumed at the capacitor is completely discharged when the power is turned on and that the power supply is "instantly" turned on (it's fast, relative to the circuit shown.)

Perhaps also you'd like the LED to gradually decline in intensity when the power is turned off. This may require some temporary storage of charge to supply current to the LEDs once the power is removed, to allow a gradual decline. Or, perhaps, you'd like an instant turn-off of the LEDs.

There are a lot of such details that may complicate a practical circuit. But each of them can be resolved, if you clearly specify the behaviors you'd like in all cases. It's just a matter of carefully detailing what you expect in all cases and then working out how to achieve those details, one by one.