Earlier quoted context omitted.
It measures the answer to the hypothetical question "If a another bank with good credit came to you right now to borrow (overnight/1W/1M/3M) in (USD/EUR/GBP/CHF/JPY) what rate would you offer them?" Which is a proxy for the banks' willingless to lend. (It had a less quoted counterpart, LIBID the London Interbank Bid Rate, where would you borrow at.)
ELI5 Why not simply measure the actual trading activity for exactly those activities the previous day?
After the LIBOR scandal, the EU brought the benchmarks regulation (BMR) which says that interest rate indexes have to be based on actual transactions, just as you say. Euribor, the equivalent of LIBOR for lending in euros, was reformed to be based on transactions:
https://www.emmi-benchmarks.eu/benchmarks/euribor/reforms/
The administrator of LIBOR proposed doing the same:
https://www.clarusft.com/rfrs-libor-is-changing/
But in the end, US and UK regulators decided just to abolish it, in favour of overnight indexes based on real transactions (SONIA for pounds, which already existed, and SOFR for dollars, which was created for this purpose).
I believe this divergence happened because of differences in the lending markets. In the euro area, there is still a lot of unsecured term lending, which is what Euribor measures. But in the UK and US, this kind of lending has largely dried up, but there is a lot of overnight lending, so they chose rates which measure that. I don't know why the euro area is different to the US and UK here. It's possible that the euro market will evolve to be more like the US and UK, in which case Euribor will stop being credible, and the euro will also move over to its overnight rate, ESTR.
Another fun quirk is that SONIA and ESTR measure unsecured overnight lending, whereas SOFR measures "repo", which is essentially lending secured with government bonds as collateral. There is a sterling overnight repo rate, RONIA, but i don't think it's used much. I think repo volumes are higher than unsecured lending volumes; if that difference gets stark enough, perhaps sterling and euro regulators will force another switch, to the repo indexes.