Lithium Ion Batteries and the 2019 Nobel Prize

Hello,

I'm researching into the 2019 Nobel Prize for chemistry, and therefore lithium ion batteries. 

Wittingham's battery was a lithium anode, a titanium disulfide cathode and a lithium perchlorate in dioxolane electrolyte.

I have always thought that batteries rely on a displacement reaction split into two halves. However in animations that I have seen show the electrons at the cathode being picked up by the very lithium ions that released them and that have just moved from the anode to the cathode. This doesn't make sense to me as if the lithium is picking up the electrons it just gave away what was the cause for this reaction to take place? I would have expected the electrons to be picked up by the electrolyte. What is happening here? 

Any help would be greatly appreciated,

Harry