'Sit it out' - Means let the time pass eg. the 12 months left on his contract.
'Walk for nothing' - Be transferred to another club free of charge.
In other words:
The France star, 23, has just 12 months left on his contract and the Gunners will not let him bide his time and be transferred to another club for free (because his contract will have run out).
Just a small point - I think 'The French star' sounds better than 'The France star'.
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