I'll take a stab at explaining from my (limited) experience. US schools receive funding from many sources, but the budget is set locally (usually once a year) by administrators and approved by a locally elected school board. When administrators make the budget they have to estimate how much money to set aside to pay substitute teachers. The administrators don't know which or how many teachers will get sick so they distribute sick hours to the teaching staff evenly. You can think of sick hours kind of like getting 'shares' in the substitute fund. Now as teachers work for a district over time these sick hours continue to accrue. Basically it means teachers who have worked there for a long amount of time and haven't needed to use the hours have hundreds of 'shares' in the substitute fund. People with a lot of accrued hours can transfer them to other employees. The amount of 'shares' in the substitute fund stays the same, but the 'owners' change. Meaning the giver loses their promise of substitute coverage, but the district can continue paying for both the sick teacher's salary AND a substitute teacher to cover their classroom, AND buy those new crayons they promised. Hope that all made sense.

This exists in Europe too. It's about company budgeting a certain total number of absent days. If someone needs to be absent longer than budgeted but it's taken from the pile of other employees, the budget is still respected. This may be easily absorbed by a big company, but not necessarily by a small local shop.

It's only common for government jobs and isn't universal there either. Many government jobs in the US receive multiple different types of leave concurrently. So for example you might accrue 12 hrs of vacation leave per month, 8 hrs of sick leave, and have a floating holiday of 8 hrs per year. If you reach a certain cap (say 100 hrs) the vacation leave rolls over into sick leave.

If you quit your job they have to pay out the vacation leave but not sick leave.

The result is that many government employees, with long careers, historically have tons of sick leave they aren't using (as in several hundred hours). When someone has a life event it's not uncommon for coworkers to donate some of their leave to the person (think having a baby or a cancer diagnosis).

You can of course take more leave then you have sick days with FMLA, but they will be unpaid (by your job, you may qualify for something like short term disability, idk). Sick leave that you have when you become eligible for pension retirement typically can apply to your retirement (potentially allowing you to retire earlier), but you still have to qualify for retirement first.

If you're 3 years from retiring and you can't qualify for early retirement with your leave, then you might have hundreds of hours you're not going to be able to use.