Would you like the short or the long answer? Haha
Figuring out a song by ear makes use of two different skills that you can develop: absolute pitch (how to tell what the notes exactly are) and relative pitch (how to tell the relationship between chords/notes). For pop songs, such as the one you posted, it isn't very crucial to know exactly what's being played, because the chords used in the song are easy to decipher and fall into predictable patterns. To break it down briefly for you, using the song "Everybody Talks" as an example:
1. Figure out the key of the song by finding the root chord. This is the chord that every other chord in the song will feel like changing to eventually. Very quickly breeze through all possible root position major chords to find one that "sits" with the song. Pop songs are easy because the root chord is normally very strong, in fact, for this song it's the first chord. And to make it easier, they hum the triad of the chord for you (0:00-0:04). This song is in Eb.
2. Know what are the "normal" chords in that particular key (in music lingo, you're looking for chords diatonic to the key). For brevity, I'll just mention that there are 7 diatonic chords in every key, numerically notated in Roman numerals: the Root or I chord, minor ii, minor iii, IV, V, minor vi, half-diminished vii, back to I. Pop songs don't want to scare listeners with weird chord progressions, so they tend to be easy and stick to every possible permutation of I, IV, V, minor vi. In the case of the song, it's Eb, Ab, Bb, Cm.
3. Find the intervals of the chord changes. Long story short, when Eb changes to Ab, there's a certain "sound" to that space between the two chords. That space will sound different between Eb to Bb, Ab to Bb, Cm to Bb, etc. At first, I can assure you the process is quite laborious; I fondly remember having scrapbooks of notes I took while figuring out chords to a song. Since a guitarist can only really play one chord at a time, it's a process of elimination to find out what the next chord is (assuming you've done Step 2). In time, you'll be able to hear I-IV, I-V movements, then extend that over longer chord changes, like I-IV-vi-V.
4. Then figure out how long each chord lasts (ie does the chord last for one measure, does it play over two? Is there half/double time chord changes?) according to the time signature. Neon Trees isn't Dream Theater, so it's strictly 4/4 for the entire song.
LONG STORY SHORT:
Intro: Repetition of the I chord. Play Eb
Verse: Repeating pattern of I-IV, each lasting two measures, so ||Eb---|Eb---|Ab---|Ab---||
Prechorus ("...everybody talks, everybody talks..."): The V chord is used to enter the chorus over four measures. |Bb---|x4
Chorus: This gets a little interesting. We normally hear progressions in groups of 4 bars. This time, it's a 6-bar group
|Eb---|Eb---|Ab---|Ab---|Bb---|Ab---|
The 2nd half reverts to a familiar 4-bar group. Then they throw in a weirder chord to spice things up, the major III, in this case G7:
|Eb---|Eb---|G7---|G7---|Ab---|Bb---|Eb---|Eb---|
Hope this helps to get you started.
