Active Voltage to Current Converter Formula
One little correction to start, V+ = V- (approximately) but not because they have high impedance but because the opamp has high gain.

You missed the part of the explanation from the linked article saying R1 = R3. R2 = R4, Rs = Rs'

Substituting those values in your circuit we can see that if the op amp inputs are at the same potential, current flow left through R1 (R3) is Vt/R1 for the negative terminal and (Vt - Vin)/R1 for the positive terminal (where Vt if the voltage at the op amp input terminals). The difference in these currents is Vin/R1. This current flows through R2 and R4.

As R2 and R4 are equal, the difference in drop across R2 and R4 is given by 
Vin / R1 * R2. As the op amp inputs are at the same potential then the difference in potential between the top of Rs' and the bottom of Rs is also equal to
Vin / R1 * R2

The difference in potential across Rs will be (Vin / R1 * R2) + IR2 * Rs'

Dividing this by the value of Rs give current in Rs = 
(Vin / R1 * R2) / Rs + IR2 = 

(Vin R2) / (Rs R1) + IR2

Iout = IRs - IR4

IR4 = IR2 - (Vin / R1)   [Because IR4 = IR3]

Iout = (Vin R2) / (Rs R1) + IR2 - IR2 + (Vin / R1)

= (Vin / R1) ( R2 / Rs  + 1)

= (Vin / R1) (R2/Rs + Rs/ Rs)

= (Vin / R1) (R2 + Rs) / Rs

= (Vin / Rs) (R2 + Rs) / R1