Help in understanding a IR repeater schematic
It looks like a simple design if you can learn to choose a driver transistor for these parts.
The design for 300mA might appear simple with these values using the model I made for this IR LED nominal values.

But getting there takes some trial and error with good NPN switches and effort in simulating datasheets.
Necessary details
You won't be able to work out the LED current until you have lots of engineering experience with diodes and transistor properties.
Once that is accomplished the solution is simply to match the transistor to the design specs and tweak the resistor values as required.
I can estimate the diode's quadratic VI properties into a linear curve over the desired linear range of current.
Although the LED can handle 1A max it is also limited to an ABSOLUTE MAX of 160mW and 85'C so choosing a safe margin is wise.  The duty cycle is expected to 50% with continuous 20~30kHz carrier and then reduced by the amplitude modulated data stream. Thus 320 mW peak might be a safe peak value and possibly with 50% 0's &amp; 1's 640mW peak.
Looking the VI chart for this LED 1.6V @ 330mA = 528mW may be a safe peak limit pending review on the data pattern.
There is always going to be some error involved with LED variances, ambient temperature etc so having good design specs is critical to this process of choosing R values, currents and transistors.  There can be a lot of errors made in assumptions.
Now digitizing the Log I vs Vf chart and converting to linear I get the following spreadsheet using a 2nd order Polynomial curve fit:

Note I computed the average of 3 Voltages and divided by 3 curre ents to estimate the tangent slope or series resistance of the diode for each current level.  Then I tweaked some graph reads to smooth the curve.
The result is a rough estimate formula for diode resistance and voltage based on a curve fit with a very small residual error.
To use the power limits above I chose from 240 to 360mA $$V_f= 1.2+I_f*5.5 Ω$$
Now the same can be done for the power transistor and I expect an Rs or Rce value around 1.5Ω which I know from experience is about a package rated for approx 1 W to achieve that low Rs value.  A PN2222A being quite low is around 4Ω might require a lower Re =0 and get too hot.
Two LEDs going to 5V simply does not make sense as the LED voltage is not 2V unless you go to 800mA peak and a lower duty cycle for power with a bigger power transistor.
hFE is also a huge variable because in order to get say 300mA I started from here.
Ve= 0.90V
Vbe=0.75V
Vb=Ve+Vbe=1.65V which means Ib= (3-1.65)/680Ω= 2mA
thus hFE=300mA/2mA=150
When Vce drops below 1 at 10% of rated current the hFE drops towards 10% of its peak current gain.  So hFE = 150 is unrealistic for a power switch and 20 to 50 is more reasonable than 10.
Here is my proof of design
Vf= 1.46V @ 230mA
Vce = 845 mV @ hFE=100
Vre = ~ 700 mV @ 230 mA across 3Ω
I won't complete the design with a transistor choice, but I just want to show you that it is not simply Ohm's Law unless you figure out the quasi-linear values for each part and choose binned high hFE value power transistors or low RdsON Nch FET which would be more expensive in high volume but much simpler for design.

This design can be easily made to work yet hard to predict Ic and standard deviation since saturating transistors have extremely wide tolerances on Rce=Vce/Ice and hFE.
Added Notes
Let me be clear, the base and collector current is not well defined in this unregulated 3V design. The transistor is well into saturation yet will never operate at the same conditions defined in the datasheet for Vce(sat) @ Ie with Ic/Ib = 10 = hFE.  It will end up with hFE being much higher than 10, yet less than the linear region with Vce=10 as defined in tables, but a nominal value may be found in figures, but not easily. So my strategy is to create a linear saturation algorithm for Vce(sat)= Vth+Ic*Rce exactly as I did for the LED in the desired quasi-linear bulk ohmic range.
After doing this for hundreds of designs, there is a relationship between the saturation resistance and Pmax for all diodes and transistors that is reliable when used over a limited range instead of the usual quadratic behaviour.
e.g. \$V_{cc}=V_{led}+V_{ce}+I_c*R_e\$
using my data for a MMBT2222A-7-F  (NPN) Vf=0.05+Ic*2Ω
\$3V= {1.2V+If*5.5Ω} + {0.05+Ic*2Ω} +Ie*3Ω \$
3V= 1.25V+ I*(5.5+2+3=10.5)   (simplifying, let I=If=Ic=Ie)    and solving  I=167mA means Re may have to be reduced from 3Ω to 2Ω for this device.
This is all from datasheets for T=25'C which won't be true if the device heats up and current will increase as Vbe and Vf reduce with rising temp.