What does the X really mean? I thought it was to mute the string but how should I do that and play another string at the same time? I hope someone can help me out with this.
While Tim and Todd's answers are correct about other types of guitar music more generally, this arrangement in particular is acoustic fingerstyle, which has other considerations for notation - this is not strictly a palm or fretting hand mute.
A distinctive element of this style of playing is the use of a range of percussive techniques to add texture to the arrangement, and these are usually marked with cross-head notes like this. Since this is a fairly niche notational requirement, there is little consensus on the best way to do this, and it is often not sensible to be consistent from piece to piece. Because of this, some transcribers will explicitly detail and comment the desired percussive action(s) - I have a Jon Gomm transcription somewhere that came with about 3 additional pages of performer's notes! However, others (as here) will leave it to the player to determine the most appropriate way to achieve the marked sound, especially in simpler pieces.
In this instance, as Bradd notes in his comment, the intention is that the player should (fairly gently) slap the marked string with the thumb to produce a percussive snare-like backbeat. Fortunately, the arranger has posted their performance on youtube where you can see this in action, particularly at 0:43.
You mute with your fretting hand. A way to mute a single string with your fretting hand is to place one finger on the string as if you are going to fret it, but don’t press with the finger to fret the note. Just keep the finger gently touching the string.
It takes some practice to fret with one finger and mute with another, but once you’ve learned it, it’s a very powerful technique.
Since it appears to be on open bottom and 5th strings, the easiest way to play the x is to palm mute them. Your picking hand is used to rest the 'heel/palm' gently on the string close to the bridge, plucking with a finger, or thumb if you prefer.(Which will give a better sound. It's a 'bass note' after all). With the 5th string x, it matters not that you palm mute the bottom string - or in fact, any others that are not to be played. It's a technique used more by guitarists who use high volumes and overdrive, which sets other strings vibrating in sympathy, so muting unplayed strings tidies up the sound.
Possibly an easier option than muting with a finger on the fretting hand. That sometimes has the disadvantage that it'll set off a harmonic. The palm, covering a larger area, has little chance of making that happen.
Bradd and birdcolour are right. Thanks for the YouTube notification! What I still didn't get was how to hit one string and pluck others at the same time, or whether this even was necessary. So I commented on the video and the arranger answered. She said that in the first part of the song the X means hitting the string with your thumb to make a snare sound, for example at 0:25 (just as birdcolour said). In this case you should simultaneously kind of flick the other strings you should play with your finger. In the last part of the song, the X means hitting the guitar body, for example at 1:07. You should then just pluck the other strings you have to play the normal way. I find both these techniques quite hard but I'll keep practicing.