Perform+preposition?

He plays skillfully on guitar.
Can you explain for me why we use the preposition “on” with “perform”?

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It is the way we express it-- We play/perform on instruments.

‘On’ = Used to indicate the object affected by actual, perceptible action: ‘The spotlight fell on the actress.’ ‘He knocked on the door.’
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Hi, MM

But we say I play the guitar, without on, don’t we?
So, I play on guitar/I play the guitar are right?

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Yes. You can play the guitar-- that is your skill. If you play on the guitar, you are probably indicating a particular event. I play the guitar very well, but last night I played on the mandolin.
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