What is the use of the line connecting drain and gate in current mirror using voltage bias?
The connection is basic to how a current mirror works.  The two gates are connected together so that both FETs are driven to the same level.  This assumes well-matched FETs, as would be the case with them being next to each other on the same IC.

The way the current mirror works is that the desired current is forced D-S thru M1.  With its gate connected to its drain, the gate will automatically be driven to whatever it takes to support the desired current.  This same voltage is then applied to the gate of M2, which in theory then allows the same current.

The purpose of M1 is to provide the right gate voltage for the desired current.  The D-G connection is the feedback that allows that, in fact requires that, to happen.