Why is an old chain unsafe?

As chains stretch, they wear the chainrings and rear cogs and the teeth on them start to become pointed and eventually don't grip the chain so well. The danger is that, when you push on the pedals, the chain slips from tooth to tooth on the chainring, which turns with much less resistance than you were expecting – you just have the friction of the chain slipping against the teeth, rather than the resistance of you driving the bike forwards. When the pedals suddenly turn much easier than you were expecting, you can very easily fall off the bike, especially if you were pushing hard, which is the time the chain is most likely to slip. Any time you fall off the bike while you're just cycling along in traffic is a potential under-a-truck moment. To some extent, you can mitigate this by cycling gently but there's always the risk that you try to pedal hard in some kind of emergency situation, forget that you can't do that, and come off.

Having said that, there is something a little confusing. Maybe you just misunderstood your mechanic, but there's no reason that you should require a new wheel, unless that's completely independent from your need for a new chain. However, you may well need new chainrings and new rear cogs: because the stretched chain wears them to match itself, they'll also skip with a new chain.

Unless you need a high-end chain (which you'd probably know about if you did) I'd expect a new chain to cost about 10 dollars/pounds/euros ("about" means the currency unit doesn't matter much among those), new rear cogs to be about 25 and new chainrings maybe 30ish. Plus labour.

Chains have a clearly defined specification for wear. Once the length of a specific number of links is longer than some maximum tolerance the chain should be replaced. If the chain is stretched, it is a good investment to replace it.

A decent chain should be around $20-$40. They are very easy to replace. No, another wheel is not necessary. You do not have to buy a new cassette. Severely worn chains will eventually wear rear cassettes and cause performance issues though, so it is a thing to consider.

Is there a safety issue? Maybe the chain will break and jam up your rear wheel causing a crash. Maybe it will break or slip and your foot will slip off causing a crash. Probably mostly a scare tactic, but not completely imaginary.

Well, if chain is worn, it might snap when you push pedal strong, which will cause your leg that was pushing to drop down very fast, and as consequence you could loose balance and fall.

That is explanation why it is unsafe. As others have pointed out it is economically unsound not to change it since it will cause damage on other more expensive parts. Something like insisting not to change oil in your car because it is expensive (and you might need to change filter with it!), or brake fluid.

So while it is great to want understand things about your bicycle, you might want to give a bit more benefit of doubt to your bike mechanic.