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.