In the process of electrolysis current can flow through a liquid because

Negative ions are attracted to the anode. In the process of electrolysis, current can flow through a liquid because negative ions are attracted to the anode.

In the process of electrolysis current can flow through a liquid because the liquid contents ions which are charged parciles (positive ions named cations, and negative ions named anions). Then, these charged particles are able to move freely through the liquid (just like electrons inside a solid conductor) which is electric current.