Current flowing inside voltage source or battery
Current is the flow of charges (many say electrons).


It's electrons in metals, but can be any other charge carriers in other media. In batteries, the charge carriers are positive and negative ions. 

When an electron arrives from the external circuit at the positive terminal and heads off into the battery, it will either release a negative ion into the electrolyte solution, or turn a positive ion arriving from the solution into a neutral species. When it neutralises a positive ion, it's getting 'absorbed by the positive charges', but of course those charges have traveled across the electrolyte. In the event of sending a negative ion off, you could see that as the electron 'hitching a ride' across the battery. Different battery chemistries may employ all of one, the other, or both polarities of ions.