Nothing....it generally does and therefore more trouble than it is worth.....dont anchor in places you believe you anchor may get fouled......I anchor everywhere I go here on the West Coast of Scotland and have never yet been fouled in the last 10 years or so!!
It doesn't if you attach the trip line to the anchor and let it go with the anchor.
Then you drop back as usual, leaving the anchor buoy (clearly marked I trust!!) as usual. We don't buoy the anchor that often, but have only ever had the odd turn around the chain that has been easily dealt with as the anchor was winched aboard. We have tried the 'tie the trip line to the anchor chain' method a few times, but have never been happy with this as we have had one or two twisting up problems.
This, though, is after having been at anchor for a few days, a week or more. Overall, I'd go for a buoyed trip line direct to the anchor as above, but definitely not let either put me off anchoring in a place I really wanted too!!
Out of interest, in 35 years of sailing, we've only fouled the anchor once (up the River Fal - but that's another story) to the extent that we had to get a diver down to recover it. So don't be put off, it doesn't actually happen very often!!!
Anchors almost always turn as the chain is paid out, so if you just let a trip-line run it is bound to end up twisted round the chain.
On those occasions when it seems prudent to use one, our standard practice is to lead the trip line over a small anchor-roller at the stern and outside the hull before hitching it to the crown of the anchor. As the anchor is lowered, tension is kept on the trip-line with a turn round a winch to prevent the twisting. That works most of the time.
The ability to use a sheet winch to bring the line back in and break out the anchor crown-first is a boon if it does become fouled.
I've never had a problem with the trip line wrapping round the chain, always used a trip in most well used harbors as there was loads of rubbish, to get snagged on. I don't bother when I can see the bottom and its a quiet anchorage,
Trip line ran over pushpit as anchor was lowered slowly to the sea bed. Anchor buoy has a pulley and weight to keep the line tight.
Almost when ever I havn't used the tripping line (in busier places) I've snagged cables, chains and old engines.