Is the greenhouse phenomena kind of like how your car warms outdoors on a sunny winter day?

Unfortunately, that's not how it works in the atmosphere.

Greenhouses trap warm air. In essence, they stop the heat from convecting away. For the atmosphere as a whole, well, there is no convection to speak of, carrying heat away from the atmosphere -- it's a vacuum out there. Further, regular soda-lime glass is just about as transparent to near IR as to visible light, e.g, the part of the curve to the right of 750 here:
Sinclair : Transmission Curves

So the story for an actual green house is that sunlight comes in, heats the air, and the hot air can't escape. The car can radiate IR just fine, it just can't convect the heat out of the interior. You'd get the same effect from (e.g.) a black-painted cardboard box -- the interior of the box is going to be warmer than the surrounding air.

You get the atmospheric greenhouse effect because some gases absorb then re-emit infrared photons. So for the atmosphere as a whole, it's all about radiation. Light passes through the atmosphere, hits the ground, most of the visible light that makes it to ground gets converted to heat (IR) and that then gets emitted upward. So the outgoing radiation is shifted toward IR. On the way out (and in, for that matter), some IR photons get absorbed by a molecule of (e.g.) C02. That molecule then re-emits that photon in a totally random direction -- including back down at the ground. The net effect is that you get infrared back-radiation from the atmosphere down toward the ground, that you would not get in the absence of greenhouse gasses, and that's what warms the earth up (more than it would in the absence of such gasses).

Must be dozens of good descriptions of this around.
Wikipedia says what I just said, about actual greenhouses:
[ame="http://en.wikipedia.org/wiki/Greenhouse_effect"]Greenhouse effect - Wikipedia, the free encyclopedia[/ame]

You can go to WattsUp, of all places, for an animated graphic:
Visualizing the “Greenhouse Effect” – Molecules and Photons | Watts Up With That?

Well, I know I've seen a nice, simple illustration of this, but I can't seem to find one at the moment.

I would not use a false analogy to explain this, no matter how appealing. There's already a lot of (denialist) misconceptions about this process out there. E.g., gosh, C02 is such as small fraction of the atmosphere that it couldn't possibly have an effect. (Answer: N2, O2, Argon are all totally transparent to IR -- as far as the greenhouse effect goes, its as if they weren't there at all. The entire greenhouse effect is due to trace gasses). And my favorite, that this violates the second law of thermodynamics (always said by people who misquote what the second law of thermodynamics says. The correct paraphrase is that the net flow of heat is from the hotter to the colder body -- they always for get the "net").

 

I'm not a physicist and I haven't taken any courses in thermodynamics, but here's my answer:

Because it's a much smaller enclosed space.

The thermal mass of the car is much smaller than the thermal mass of the Earth, so an external radiant heat source can raise the temperature of the thermal mass inside the car and that raises the inside air temperature much faster than it can do with the Earth as a whole. Just like you can boil a cup of water a lot faster than a 5 gallon barrel from the same heat source. (Remember, heat can be transferred by radiation, convection and conduction.)

Similar to what Chogan says, the heat can't escape from the car (convection is stopped) but I think he's forgetting about the glass windows allowing in radiant heat, same as what happens in a green house (but not a black box, where only the surface is directly heated, not the interior).

So this is the same principle as a greenhouse. It gets warmed because the air is captured by the glass windows. Not because of gases in the greenhouse, just so you know. The gases cause a "greenhouse-like effect", which is then shortened in common vernacular to greenhouse gases. In your case, it could be "car-like effect", so we could also call them sunny car gases...?

So the greenhouse gases (GHG) (or SCG ) allow light to come in, but then absorb radiant heat in the infrared spectrum coming back from the earth. They keep this heat instead of allowing it to bounce back into space, so the atmosphere warms up. It keeps the heat close to earth instead of allowing it to radiate into space, so in that sense it acts like the glass windows of a greenhouse or a car (which stop convection and conduction instead of radiation, so the greenhouse analogy is imperfect to begin with).

History of the discovery of greenhouse effect: The Carbon Dioxide Greenhouse Effect
The first two paragraphs of this page are useful: NASA GISS: Science Briefs: Greenhouse Gases: Refining the Role of Carbon Dioxide

 

I think the boat has long ago sailed on the analogy. And at a basic level it works, light comes in and is trapped as heat. I seriously doubt anyone needing an explanation on the greenhouse effect will be bothered by the faults in the analogy (as an analogy).

'Smaller space' has nothing to do with it. 'Enclosed' is a bit tricky, as in some ways the Earth is enclosed in that no heat convects away from it, same as a greenhouse or car. Heat radiates just fine from a black box. It should also be noted that simple soda lime glass is NOT what is in car windshields.