How do we have a fixed current in this case?
It's only relatively fixed. If the voltage V1 is something like 5V then the current will be around 4.3/R1 under typical conditions and will only change a bit with temperature or for different  transistor characteristics. For example it would change by +0.05% per degree C.  

The matching of the transistors is another matter- they must be very closely matched - that circuit is pretty much useless unless the transistors are on the same die so that they are both matched and at the same temperature.