What causes ions inside of a lithium ion battery to move between the cathode and the anode?
Difference in tendency of chemical elements to gather or to give eletrons, called the reduction potential
(I think those differences are not easy to explain exactly because there is much quantum mechanics involved; there are rules of thumb you may remeber from chemistry class about elements being more or less noble or unwilling to give some of their electron) together with
electrodynamics and
thermodynamics


are underlying principles involved.