Really old cars have very simple electronics. Even after computers started showing up in cars, they were pretty simple and didn't interact much with the security aspect of things. When you start a car like this, you're just making a connection between two wires to power up the electronics, and briefly making a connection between another two wires to run the starter motor. The only security in the whole system is provided by the fact that the connection is made by a switch that requires a key to turn it. If you don't have the right key, you can't turn the switch, and that means you can't connect the wires.
The trouble is that the wires must be fairly exposed to the occupants of the car, since the switch has to be accessible. That means you can just bypass the switch entirely by removing the appropriate covers and attacking the wires directly. This is "hotwiring."
Physical locks are also not all that difficult to defeat directly. You can pick an ignition switch much like you might pick any other lock.
Starting around the late 90s or so, car manufacturers started adding more robust security measures. These include things simple like locking the steering column when the ignition switch is off (thus preventing you from driving the car after hotwiring it), all the way up to authenticating the key with a relatively sophisticated protocol, and having the engine computer refuse to run the car unless it can sense a real key.
As a result of these changes, the list of most stolen car models is still topped by cars manufactured in the late 90s. Low-end Hondas from around 1998 are right at the top of the list, because they occupy a sweet spot of being relatively valuable and still fairly easy to steal. Modern cars are stolen literally orders of magnitude less frequently; about 100,000 older Hondas stolen per year in the US, whereas new cars are stolen at a rate of hundreds per model per year at worst. Also as a natural result of these changes, car theft is way down in the US. About 700,000 cars were stolen in the US in 2013, compared to almost 1.7 million in 1991. Pretty much the only way to steal a newer car is to either tow it away or steal the owner's keys. (A common scenario for car thefts is a burglary turned into auto theft when the burglars find car keys in the house.)
It's easy to get into many older cars. Slim jim past the window is the classic example (and I opened my 80s Toyota with a coathanger multiple times when I locked myself out), but many times the locks could be opened by keys to other cards from the same manufacturer as well, they just didn't seem to be that precise. And of course, smash the window as a last resort, that wouldn't set off an alarm in the past. Nowdays cars have recessed lock things in the door panels (or button-controlled-locks that can't be as easily manipulated with a coathanger, or even that don't work at all if the car was locked from outside) to help prevent this, and the interior of the doors has more protection built around the lock mechanism so you can't easily fish through there and hook onto the right lever.
Once inside an old car, starting it is usually just a matter of shorting the right pair of wires. Or using brute strength to turn the ignition cylinder even if they key isn't an exact match (or maybe with a screwdriver, as another poster mentioned doing in the past in this thread). Modern cars have chips in the keys so that it's not just a matter of closing a circuit, the key has to be coded to the car.
Or just tow the car somewhere and work on picking the lock later at your leisure. Overkill for a common car, but for something really nice it could be practical. Nowdays your more expensive cars have tilt and motion sensors that'll set off the alarm if you locked it, left it, and someone else comes up and tries to tow it. Possibly GPS tracking or similar as well, IIRC, on some fancy stuff.
The fob-in-pocket entry/pushbutton start stuff gives up some of those improvements given an exploit like this, but overall I'd say is still much more secure. You need specialized hardware (that's only useful for breaking into someone else's car) and it wouldn't work to, say, steal cars from an airport parking lot or somewhere else where they were left and the owner wasn't in range. Keeping your car in a garage at home seems to mitigate a lot of the easiest vectors for this attack.
To add to the comments about immobilisers, in a number of countries (UK and Germany, amongst others), from 1998 all new cars were required to have an engine immobiliser. Most manufacturers simply made them standard for all countries, so nearly all cars built since 1998 have had them fitted.
The trouble is that the wires must be fairly exposed to the occupants of the car, since the switch has to be accessible. That means you can just bypass the switch entirely by removing the appropriate covers and attacking the wires directly. This is "hotwiring."
Physical locks are also not all that difficult to defeat directly. You can pick an ignition switch much like you might pick any other lock.
Starting around the late 90s or so, car manufacturers started adding more robust security measures. These include things simple like locking the steering column when the ignition switch is off (thus preventing you from driving the car after hotwiring it), all the way up to authenticating the key with a relatively sophisticated protocol, and having the engine computer refuse to run the car unless it can sense a real key.
As a result of these changes, the list of most stolen car models is still topped by cars manufactured in the late 90s. Low-end Hondas from around 1998 are right at the top of the list, because they occupy a sweet spot of being relatively valuable and still fairly easy to steal. Modern cars are stolen literally orders of magnitude less frequently; about 100,000 older Hondas stolen per year in the US, whereas new cars are stolen at a rate of hundreds per model per year at worst. Also as a natural result of these changes, car theft is way down in the US. About 700,000 cars were stolen in the US in 2013, compared to almost 1.7 million in 1991. Pretty much the only way to steal a newer car is to either tow it away or steal the owner's keys. (A common scenario for car thefts is a burglary turned into auto theft when the burglars find car keys in the house.)