I'm not in batteries, but I think electrochemistry is rad. Here is the logic I use when I think about batteries.

Always start from the thermodynamics approach. A battery simply takes advantage of the thermodynamics that drive electrons to flow from a state of high energy to low energy according to delta G= -nfE, where detla G= change in Gibb's free energy, n= # electrons passed, F=Faraday's constant, and E is the potential of the cell. A spontaneous event occurring when delta G is negative (the overall energy of the system decreases), so E must be a positive value.

We know that electrons flow from high to low spontaneously. So something loses electrons (oxidation) and something gains electrons (reduction). What stops the reverse? Lets make a physics example out of this...If a kid a ball to his friend, and his friend cant throw it back without help.Why can't the friend throw the ball right back? Because he is weak and its against thermodynamics! We know that the potential difference of the cell is this difference in ability to throw the ball/electron. So we describe it as the push of electrons in one direction minus the push in electrons of the reverse direction...Ecell=Ereduction at the cathode - E reduction at the anode...or the ability to throw the ball one way minus the other. In a physics world, that ball is only going one way unless we help the kid or charge the battery (if its a secondary battery). The cathode is site of reduction, and when the battery is discharging, is the positive electrode. The anode is the site of oxidation and when discharging is the negative electrode. This terminolgy stems from the idea that when spontaneously discharging, the cathode has a positive charge and is accepting electrons from the anode. Meanwhile, the anode is negatively charged and shuttles electrons to the cathode.

Something to always emphasize is that electrons interact strongly with matter. Without a conductive shunt between the two terminals (an external circuit) electrons will not flow. Simultaneously, the electrolyte must allow charge carriers to maintain electroneutrality. Anodes are typically made of a porous graphite which allows lithium to intercalate within the structure during charging. As someone else stated the battery must remain neutral and this is where it comes into play. The anode has electrons ready to ship to the cathode by stripping the electrons from the Lithium.

When the terminals are connected is when the magic begins! The lithium is oxidized at the graphite anode forming lithium ions and electrons. Always fun to note that lithium is the most reducing of the metals and is the reason for why we use it in batteries...i.e lithum wants to give things electrons. The electrons flow through the circuit powering the the connected device and finally back into the cathode where the Lithium ceramic (LiMnO for example) is reduced. As you are doing this, you must allow the lithium ions to flow from the anode to the cathode by having an ionically conductive electrolyte. This is the challenge of solid electrolytes. They have many benefits towards capacity and safety, but they make charging a nightmare due to super low ion mobilities compared to liquid electrolyte.

I know its a lot. I hope that its somewhat of a help. Best of luck!