They typically will make a temporary connection with a nearby device (usually Apple or Android) and let them know they are nearby. Their identification is anonymous though. The device they connected to can then connect to Apple's servers and provide the identification along with the geo location of the device performing the upload. If the device has a GPS with a position, that's what will be sent.
But if the device was just turned on or is in a location with poor GPS signal, the accuracy of the location can be poor. Due to the nature of GPS, the precision can be much higher in one direction and lower in another, so it may be +/- 20 feet north/south, and +/- 100 feet east/west. If GPS is unavailable but the device has a cellular signal, it may instead try to estimate the location using nearby tower strength. (the phone knows the GPS coordinates of the towers) This can also be low precision if cellular reception is poor or there are few towers nearby with a decent signal. The same +/- variations that happen with poor GPS can happen with poor cellular.
GPS DOP (degree of precision) is available to your GPS and some will display the uncertainty, but most cell phones don't display their DOP, for their GPS or for their cellular. Unfortunately, they're happy to provide an exact GPS coordinate and say "this is where I am", without any warning of just how good or bad the precision actually is. (my GPS will give me a rough indication of its current precision, sometimes while showing me driving down the road that's one block to my left or right...) DOP can be included with the report, but again whatever is interpreting it needs to inform the user of the uncertainty. I don't think "find my device" displays the DOP of the last position.
If it puts a thumbtack on a house, but tells you the average DOP is 300 feet, you can't just go busting down that door. It sounds like that's what happened though, and for that they ought to get held accountable.
And making matters worse, this is all reporting the position of the device, not the AirPod, so you have to figure in how far away the AirPod is from the device, and further ADD to the DOP, which is just going to make the reading even less accurate. Even if the phone had excellent DOP of say +/- 3 feet, if they're driving down the street and pass by the house, there's NO way to tell even which side of the street it's on, since the DOP from the AirPod to the phone is going to be something like +/- 40 feet to reach to the AirPod in the house. (so the final DOP is +/- 43 feet) And so realistically, that position is going to cover at least half a dozen houses.
But from what I recall of Find My Device, I think a lot of the blame here is on Apple, for not being clear about the accuracy of the indicator. The last time I used it, it pulled up a map and put a red pin where it thought my laptop was, and gave NO indication of the blob-shaped area my phone was LIKELY to be in. It just put a pin on the map and said "here it is!" But it was in my house, I had a good signal on my phone, and the computer was in the same room, so both the phone and the computer had good DOP. And this is probably the same scenario that people were in when they turned it on and played with it for the first time. "The pin is on my house and that's where my computer is, this seems to work really well!" This leads to an unreasonably high confidence in the precision of the report.
It's also important to remember that this is the LAST REPORTED position. If that AirPod was in a car that was following another car, whose driver's phone reported it, and it hasn't checked in with any other device since then, it could easily be MILES away from that reported position now. You've got to pay attention to the exact timestamp of the report, and again that may be something the swat team didn't do.