Because I think, the conversion would be done even without this transistor.
The opamp will set a voltage based on the inputs, not a current - it's a normal opamp by the looks of the schematic symbol, not an operational transconductance amplifier (OTA) which would set a current based on the inputs.
Also, the amount of current an opamp can sink or source is generally very small, so even an OTA without an external 'buffer' like the MOSFET circuit would have extremely limited V-to-I range.
If this still doesn't make sense to you, please explain why you think the conversion would be done without a transistor.
Think of the circuit this way. Assume that your Vin signal is zero, the output of the opamp is zero and because of this, the signal on the gate of the MOSFET is zero, the MOSFET isn't conducting and subsequently the signal on the inverting input of the MOSFET is zero.
Assume that the Vin signal goes to 1V. There's now a difference of 1V between the op-amp inputs. The opamp output will start slewing towards the positive rail, since the non-inverting input is higher than the inverting input, and since the MOSFET is off, the opamp is open-loop with extremely high gain. Eventually, the opamp output voltage will reach the gate-to-source threshold of the MOSFET, and it will start conducting.
One of a few things could happen now.
If the off-page connection to the drain of the MOSFET goes to a voltage source, the MOSFET will start controlling the current that flows through it as a function of the gate voltage. The current through the MOSFET creates a voltage drop across R1. The voltage across R1 is the feedback - we're no longer open-loop - since the R1 voltage gets fed back to the non-inverting input. The system will reach equilibrium when enough opamp output voltage is generated to control the MOSFET to allow exactly enough current to flow through R1 to create an identical voltage drop to Vin, and will maintain equilibrium by adjusting the opamp output as Vin (or the MOSFET dynamic resistance) changes.
If the off-page connection isn't connected to a voltage source, no current will flow through R1, the opamp will remain open-loop and the opamp output voltage will rail to its maximum possible positive output. The MOSFET will be on, but not doing anything.
The advantage of this approach is that a small, relatively 'weak' opamp (in terms of drive capability) can be used to control tens, hundreds, even thousands of amperes - it's just a matter of the size of the MOSFET and the power handling capability of the sense resistor.