What does the inside of a large lithium ion battery look like?
It's not safe at all to open lithium ion (or lithium polymer) batteries. They contain volatile, flammable and possibly toxic materials. Opening them is at least a major burn and fire hazard. They can also spontaneously dump their stored energy if you damage them, which causes the battery to get hot and possibly ignite or explode.
Cylindrical Lithium-ion batteries generally consist of a roll of a stack of: a copper anode foil that's coated with black or brown ion absorbing material, a thin white-ish separator made out of a polymer (plastic) that can let lithium ions through but physically separates the electrodes, and an aluminium cathode foil that is coated just like the anode. This roll of electrodes and separator is soaked in liquid electrolyte, which is usually a clear solvent-based liquid with dissolved lithium salts and other additives. It evaporates very quickly and is also highly flammable. The anode and cathode are connected to the battery's terminals via metal strips and the entire thing is tightly packed into a metal tube (the casing).
Flat pouch-type lithium polymer batteries contain the same components (anode, cathode, separator, electrolyte). They're just folded instead of rolled to make a flat stack of alternating electrode and separator layers. This stack is then wrapped in a thick metal foil to provide at least some mechanical protection and to prevent the electrolyte from evaporating.
Larger batteries (like in a car) are made by combining lots (possibly thousands) of those smaller cylindrical or pouch-type cells. A bunch of cells is connected in parallel to provide higher capacity and current-handling capability. Multiple of these &quot;chunks&quot; of paralleled cells are then connected in series to provide the required voltage to the device that has to be powered. For example, a &quot;3S2P&quot; battery is a series connection of three &quot;chunks&quot; of two paralleled cells each. This gives twice the capacity of a single cell at three times the voltage, for a total of six times as much energy (since you have six cells in total). This is a very popular configuration for laptop batteries.
Here's a video of someone disassembling an 18650 lithium cell. Don't try this at home!
To answer the additional question that you edited in: Separator breakdown either happens due to mechanical damage to the cell (i.e. someone whacking, slicing or crushing one of those pouch-type cells) or because of electrical stress. Overcharging and over-discharging a lithium-ion cell can cause metallic lithium dendrites to grow within the cell which can puncture the thin separator and cause a short-circuit, which in turn results in the cell rapidly releasing all of its stored energy as heat. This will then cause the cell to puff up, vent or ignite.