This is the example sentence: 私の趣味はギターを弾くことです。
My translation - Playing the guitar is my hobby.
However, why is it not: '私の趣味はギターを弾きます。’
Basically I don't understand this use of 'ことです'.
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Short answer: nominalization.
In this case, it's not really a quirk of the Japanese language, at least you're doing pretty much the same in English as well.
In English, we don't say
*My hobby is play the guitar.
*As for my hobby, play the guitar.
A is B needs two things (either a noun, or mentioning a word or phrase, as in swim is a verb) for A and B. So we use the gerund
playing, which basically refers to the act of doing the action
My hobby is playing the guitar.
This does not mean that playing the guitar is done by your hobby,
playing the guitar is behaving like a noun here.
The same goes for Japanese.
私の趣味はギターを弾く could perhaps mean
my hobby plays the guitar, but what you want to say is that the act of playing is your hobby.
弾く事【こと】 (こと is usually written in kana) is almost literally the the act of playing.
playing the guitar, which is your hobby, ie. 趣味.