I presume it means you have to collect your bags and take them to the check in desk to check them back in for the second flight, rather than the airline transferring them from one plane to the other directly. Starting from the UK this probably doesn't add that much time to the connection compared to connecting airside, given you still have to go through the same passport control and security in both cases. If the UK were a full part of Schengen then it would be a much bigger penalty as you'd be adding an extra security screening check that you wouldn't otherwise need to undergo.
It depends on the airline's policy.
Basically, your bags are the responsibility of the first airline you check in with. By IATA rules, they are only obliged to check your luggage in as far as you are ticketed in the PNR for the flight you're checking in with (or less, if there are specific rules about re-checking for customs at international to domestic connections). They can check it through as far as you like out of goodwill - and some will do this for the same airline and alliance/codeshare partners - but most won't, these days. Since the advent of e-ticketing, the cost to airlines of checking bags through has been easily identified, and it's one they'd rather avoid if you haven't paid them for it.
The same applies for accommodating you in case of misconnections, whether due to late arrival of an incoming flight, or delays at immigration or baggage reclaim. Airlines are not liable for consequential loss, and so are at no obligation to reaccommodate you if you miss a split flight. Again, they may do so out of goodwill (or if you have a flexible ticket - although then, it's subject to capacity) for same-carrier or alliance partner flights, but this, again, is getting rarer.
Most (but not all) low-cost airlines will not issue any connecting tickets, so the above is always the case for them.
Another technique for saving on flights is to use a connecting, initial flight into a hub, where you would have flown from the hub directly. For example, London to Toronto on BA is generally more expensive than, say, Milan - London - Toronto on BA. Now, bear in mind that if you miss any sector of a flight, all subsequent sectors are cancelled, so you do need to physically start in Milan. However, book a stopover, make a weekend of it, and you may find a nicely discounted ticket with a cheap weekend away into the bargain. If you have time on your hands, you can do the connection in London the same day and save on the departure tax, too (departure tax is not payable on connections of under 24 hours).
The other point to remember is that by splitting a ticket, you may have to enter that country in order to connect. While that doesn't really matter for an EU national on an EU connection, it can get quite complicated. Plus, somewhere like Schiphol (mentioned above) is designed for connecting - you just walk from gate to gate. Having to go out and in again is a significant detour.
Personally, I've done the split flight thing a few times, although in my case for air miles redemptions, where I can save on the taxes, fees and fuel surcharges that you have to pay. The most recent example was getting a very cheap Air Berlin flight to Duesseldorf (I think it was about £20 all-in) with a Rail+Fly ticket. Had a potter about there and Cologne on the ICE trains, left from Frankfurt in the evening. Saved about £150 by doing so, but there was a gap of many, many hours between the flights.
Train connections do work too - you can get particularly cheap tickets to Disneyland Paris on Eurostar connections via Lille (not the direct train!) and all of these trains stop at Roissy-CDG. Again, though, you need to leave contingency time, as the connection isn't guaranteed. Dublin is another good departure point for cheap taxes - and it has good connecting flights to the USA and Canada (with immigration pre-clearance to the USA, which is a great benefit).
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Does anyone know what "not connecting" actually means in this context?
A connection is a flight booked on the same PNR (passenger name record), with a gap of under 24 hours between arrival and departure.
As mentioned above, you cannot assume a non-connecting flight will be honoured in case of a misconnect, even with the same airline - and not just the low-cost airlines, either.
Worse still, if the second portion of the flight is a return flight and/or has another connection, you will be treated as a no-show, and all subsequent sectors can be cancelled according to ticket conditions. If it's a non-flexible ticket, you get nothing back (well, you may get taxes back, less a hefty admin fee...)
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I wouldn't be shocked if Air France, for example, would sell 2 "unconnected" tickets through Paris, but would allow you catch the next reasonable flight out of Paris for a small fee much less than the APD should you miss the "unconnected" flight ....
It's not in classic airlines' interests to do so. They make most of their money off full-fare tickets from (usually) business travellers. If they allow people on flexible tickets to change tickets at a far smaller fee, they're eating into their most profitable area.
easyJet is a rare example of a European airline that will allow same-day changes/standby at a fee. Even then, it's restricted.
US airlines tend to do same day standby/changes at a fee more readily, but again, these are subject to many restrictions, such as a 3.5h limit. Which is fine if your destination has a flight every few hours, but if it doesn't, you're SOL.
