"Breaking your back" means that one or more of the bones in your spine (which helps you stand up straight and move around) has been damaged or broken. When you over-flex or bend too much, it can put a lot of pressure on these bones and they might break. It's like if you bend a pencil too much, it might snap. The soft tissues in your spine, like muscles and tendons, help hold the bones together and keep them in place, but they might not be strong enough to stop the bones from breaking if you bend too much.

That still doesn't quite add up in my head - when I bend a pencil, I'm over-flexing a single, rigid structure. The spine is lots of smaller rigid structures, held together by soft, bendy stuff.

"Breaking your back" often isn't used in it's strictest medical definition, so doesn't necessarily mean breaking a bone. It can mean dislocating one of the bones/joints to the point that it affects or damages the spinal cord. That, in layman's terms, means "something in your back is broken" (aka "spinal injury"), which loosely translates to "breaking your back". Same way a "broken" ankle or wrist can actually just be a dislocated bone. It's just easier/more common to say "It's broken" for most people who don't need an exact medical breakdown/explanation to get their point across

You're assuming that the soft, flexy tissue will break before the bone will. But the tissue, being soft and flexy, specifically won't break, whereas the bone, being rigid, cannot flex at all. Both materials being subjected to the same load, it's not a given that the tissue will break before the bone will.

Not quite - I'm assuming that soft tissue will bend/flex before the bone breaks - in fact, it must to allow what spinal flex we have as it is.

If you pull on your hamstring too hard due to flexing joints around it too much, the hamstring tears rather than your femur breaking or similar. This is different to how the spine and its connective tissue/muscles works (in that it seems it's the bones that come apart, rather than anything softer), and it's the explanation of that difference that I'm looking for.

At no point in flexing your joints are you putting any significant stress on your femur or other straight-bone in the limb.

The tendons and ligaments are providing support in other directions/axes that your bone isn't providing. If you twist an ankle, your tibia or fibula isn't bearing the major stress; you could twist someone's ankle 360 degrees around, completely destroying the entire tendon structure, and the tibia/fibula will be structurally intact. Likewise, if someone hits you in the shin with a 2x4, your ACL or Achilles isn't bearing the major stress; completely smash the tibia and fibula, and the ACL and Achilles, while now useless, won't tear.

In your back, the discs and vertebra are basically in-line. They are sharing the load between the two whenever you're bending the spine. So that force gets distributed across/applied to both, which means either can break from the same load.

The most common types of fractures (IE Breaking Your Back) are spinous-process fractures and compression fractures.

The spinous process is the little "dinosaur bone" that sticks straight out the back. There are numerous muscle groups, ligaments, and tendons that "tie" these together and when you go into full flexion, they pull on each other. With enough force, the tip of the spinous process can snap off.

Compression fractures happen when the vertebral body (the big round part) is literally compressed. The disc will not rupture because it's literally designed to absorb and distribute compressive forces.

You just have to remember that, generally speaking, the way the spine fractures is very different than a long-bone.