I’ve been watching the live performances from the Cavern Club online and most of the musicians play an acoustic guitar.
Why do they sometimes use a capo? It can’t be because it makes the chords easier to play, right? Sometimes one of the artists will use the capo on one song and then the next one they won’t, but they’re all over the neck of the guitar playing various chords.
It changes the pitch of the otherwise open strings, and the sound/timbre, and the voicings of chords. There are even weird capos that press down on different frets on each string. Capos are not always used in all circumstances; sometimes musicians have to retune the guitar.
Here’s an example that applies to my playing. I’m most comfortable playing a lot of blues songs in E, but G may fit better with my vocal range. I can play “Key to the Highway” better in E, especially when I’m going all over the neck. However, I’m most comfortable singing it in G. I just slide the capo down to the 3rd fret, and both my vocal chords and fingers are happy.
A capo (unless it’s a weird one that only presses on certain strings, as @DPRK describes) changes the tuning of all six strings (and, thus, the key in which the guitar is tuned) without having to re-tune the strings. it takes only a few moments to put a capo on or off, as opposed to, potentially, several minutes to properly retune all six strings.
As @P-man notes, sometimes it’s used by guitarists to place the song in a key that better fits the singer’s vocal range. Other times, the change of key allows the guitarist to execute chords that are more difficult (or even impossible) in a guitar that’s in standard tuning.
That’s why I do it. A song that’s written in E FLAT? Those are some awkward chords… but capo up one fret and play D chords, easy peasy!
(So, just mentioning the I, IV, V chords, I’d be playing D, G, A instead of E flat, A flat, B flat)
Ooh, I just had a thought: when you’re doing a bar chord, your index finger is essentially a capo. For an F chord, you’re doing that same “Capo up a fret and play an E chord”!
Here is a Youtube guy demonstrating basic use of a capo. He is presumably advertising his guitar course for complete beginners and from that perspective he indeed describes the uses already mentioned in this thread: (1) transposing a song up to make it easier to sing; (2) substituting for your finger when playing barre chords; (3) what he calls the most important reason: changing awkward chord shapes to different shapes to complement certain styles of music.
To digress a bit, not directly related to capi or even to the guitar, but when chords are indicated in sheet music, for example, it might say G, musicians do not always play the same pitches (or even stay on the same “chord”)— it all has to fit the relevant style.
Huh, I know little about guitar playing, but these are not the answers I was expecting to see. I can understand that a beginner or even an advanced amateur would use a capo to make certain chords easier to play, but I had assumed (and reading between the lines, I think the OP assumed) that a professional musician would be able to fairly effortlessly play any chord. I’m surprised to hear that they use a capo to avoid difficult or unfamiliar finger positions. Worst case, as @digs noted, you can use your index finger in a bar chord to do exactly what a capo does. Maybe I’m underestimating how difficult some chords are, and they’re hard for even the most skilled musicians?
My undestanding is that there are some chords that are either impossible, or very awkward, to execute as a barre chord, and even professional guitarists use capos. As this article notes:
It doesn’t matter how good you are at guitar. Different chord voicings sound different and there will be different possible chord voicings with a capo for a given key.
Also, changing fingering takes some time no matter how fast you are. Less left hand movement means a smoother sound. It’s not just forming the chords, it’s the transition between chords.
For example, imagine playing an open C to an open D chord. You can play those same shapes as barre chords but there’s always going to a perceptible sound difference if you are trying to smoothly and quickly change between those chords in barre form.
Richard Thompson with a bit of brand endorsement:
Richard Thompson using one - I think we can qualify him as a professional 
:
Definitely some pro technique: I guess he has re-tuned his strings AND uses a capo on top of that? Anyway, for example you can clearly see him repeatedly play B-flat (or whatever it is) on the second string. Without a capo that would be impossible without using a finger; the way he is playing would be physically impossible without the capo.
There’s two things happening with a capo being conflated: 1) Changing the key of a song by moving the nut up and playing the same fingerings; 2) playing your “fixed key” song, that you need to play and sing in a specific key, with different chord shapes, maybe folkier etc.
A capo moves the nut of the guitar higher. The strings are shorter, and hence a little more brittle, and defined, the higher up you go. Any altered “technique” of fingerings is only affected by the tension and width of the neck at the place where you are playing.
You can do it to sing in a better key for your voice. But for acoustic songs a capo can just be a great sound. You don’t use it on a les paul to play hard rock.
Pros may get fancy and use a capo on some strings and not others. But other than that I can’t think of “technique” or expertise that a capo gives access to for a player other than changing keys, or chord shapes.
Replacements - Here Comes a Regular
On one of my guitars, which I like playing a lot, the strings are a little higher off the neck than is comfortable, but when I capo it, it changes the angles of the strings a little and makes it easier to play.
I’d say take it in for a set up. It probably doesn’t have to be a problem.
One common technique in recording studios is to overdub a strummed rhythm guitar part played with open chords at the nut with the same chords played in other positions higher up the neck. A capo is often used for this, because it means that the overdubbed chords get the same “open note ringing out” sound as the first set of chords, meaning they blend together nicely. You could play the higher part with barre chords of course, but you would get a different effect.
An anecdote: I once saw Neil and Tim Finn (of Crowded House fame) play a smallish gig to promote their album Everyone is Here. They launched into the first song of the night, to discover that they had their capos on different frets - they were one semitone out from each other. Ouch. 
What the OP might be missing is that there are lots of ways to play, say a G-major chord, and they use a different subset of the possible notes that could make up a G-major chord over two to three octaves. So even just strumming, one way of playing a chord will sound a bit different from another way of playing the same general chord. And some ways of playing it might need open strings, so you can only play it with a capo at the right spot.
Then, when finger-picking, the exact form of the chord is even more important; you can’t fingerpick a barred G major the same way you’d fingerpick an open-string G-major. So if you need to play a song you would fingerpick in G up a whole step, you can’t just play A chords on the regular guitar; you need to capo up a step, so you’re playing the shape of a G but getting an A.
Thanks for that!
(But now I want to say ‘cappo’…)
Also, even among “professional musicians,” there is a rnage of human physiology. Some people have fat fingers or short fingers, or small hands. That can make some barred chord shapes, for example, impossible for that particular musician.
About a year or so ago, I saw a Nashville-based singer/songwriter/guitarist play live and I noticed that he went out of his way to avoid bar chords and afterwards I asked him about it and he said something to the effect of that the goal was just to get a good sound rather than to try to show off that you can make your hand and fingers do every difficultthing.
Barre chords are blocky and chunky and less well articulated than a pro will want. Many times pros play chords up the neck without a barre, by leaving out (higher and/or lower) strings. Especially rock players, wtih those power chords. Jazz players need more articulation.
Barre chords are really a good way to picture and keep track of the spacing and intervals between chords of a song.