Hi,
A V to I converter like this kind is a relatively simple idea...take a voltage and convert it to a current so that when you vary the input voltage the output current follows that variation. You dont need a transistor if the output current does not have to be higher than what the op amp can put out.
In fact, you dont even need an op amp if your load is a certain constant resistance because when you apply a voltage across a resistor a current develops.
If your V to I was 1 to 1, you would input 10mv and get 10ma output, 100mv in and 100ma out, etc., although many of these circuits use a different ratio of input to output. In your circuit, when a voltage of 1v is applied to the non inverting input terminal of the second op amp section in order to get 1v at the inverting terminal 1v would have to appear across the emitter resistor and that would mean 1/100 amps (which is 10ma) would have to flow through the emitter, and ignoring the base current (high gain transistor) the current in the collector is roughly the same and so the current in the collector is 10ma too.
Ignoring the base current means there will be a slight inaccuracy in the current setting (about 1 part in 100) but that is often good enough. In this case of driving LEDs, it should certainly be good enough. For higher accuracy however a slightly different circuit topology is employed (where the current in the collector is measured rather than in the emitter as in this circuit) making the output setting much more accurate. Sometimes the load is placed in the emitter circuit which also provides for higher accuracy.
Note also that this circuit is a current 'sink'. Placing the load in the emitter circuit would make it a current 'source', although for LEDs it doesnt matter that much.
In terms of spice or general circuit analysis, this would be called a "voltage controlled current source", or simply an "I of V" dependent source.