How can I get that out of my recordings?

1) Mic placement - you can experiment with this now
2) Mic type/model - if you have different mics (LDC, SDC, etc). I don't know what you have or your ability to get more, so I put this second.
3) EQ after the fact - but only to fine tune after you have gotten the sound you want via the above.
4) Compression - easy to abuse, so I put this last

As for the specifics, you'll need to wait for some of the more experienced folks to tell you the details of what usually works, but in the meantime, try working on these (especially #1) yourself - best way to learn.

You should also note that in the example you posted, there are 2 separate guitar tracks (panned L and R), each with probably different mics, placement, eq, comp, etc. So while just one of them might sound "off" (too much this, not enough that), together they make a nice result.

Alot of that is just coloration of the mic being used. It's real "picky", besides all the added effects.
Like this:
http://www.studioauditions.com/audio/Products/Microphones/Audio_Technica/Audio_Technica_AT3060/Audio_Technica_AT3060_Acoustic_Guitar.mp3
Or this:
http://www.studioauditions.com/audio/Products/Microphones/CAD/CAD_e100/CAD_e100_Acoustic_Guitar.mp3

You won't get that by just sticking the mic up by the player's hand, but placement is a huge part of it. I'm right now trying for a picky, thumpy sound with my SM57 (the only mic to my name), and I can kind of fanagle it into this kind of sound after about five hours in front of my computer messing around with effects, and about an hour spent placing the mic.

Another thing is the chords he's playing sound like where you're playing a bar chord, but you leave your index finger kinda loose, and mute all those stings, so all you hear clearly are the other notes you are fingering. Somethin like that.

Besides that, I'm not really sure what kind of effects he's using. You'll have to wait for someone with more experience to get an answer there.

You guy's got it right.

Some other important considerations to achieve 'that' sound.

A good instrument - in tune, clean, with new strings.
A good preamp - the Earthworks 1022 was used on the Studio Auditions tracks, that's a nice transparent pre.
And you gotta have a good room to play in!!!

8-)

+1 on new strings!

I doubt that the mic placement was right at the strumming hand. This usually leads to too much noise and can distract from the track. There is a nice tight sounding mic being used and a decent pre. I dont think its really really high-end stuff as there isnt a lot of width and depth to the 10 secs. I heard. I would say the instrument in this case is playing the biggest part of the equation.

That's nothing special in terms of recording. What I hear on that excerpt is a consistent player on a good instrument fitted with new light-gauge strings. He's playing to the mic and not trying to make too much sound. I think the mic may be an LDC, and it's positioned half way up the fretboard to avoid the zing from the pick on the strings being swamped by boom from the soundhole. There may be some DI pickup blended in, and possibly a second mic at a distance to capture the space.

I assume this is an intro to a band track and not a solo guitar recording. The mic choice, mic placement, preamp choice and maybe whatever EQ used have all been done to get a sound that will sit in a mix with a vocal line and other instruments.