This is the context:
D’autres régions sont propices aux loyers onéreux : les destinations de vacances. Et les Grisons remportent la palme, loin devant le Valais et le Jura. Les 2000 francs de loyer y sont fréquents en de nombreux endroits.
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Literally, remporter la palme is to win a trophy.
It is also an idiom to designate a “winner” (quotation marks included). Very often it's used sarcastically, then it would of course mean “the worse”.
In your sentence it is not completely clear whether having high-priced rents is considered good or bad, it depends on the context. However it is clear that Les Grisons have the highest rates.
In that sentence "remporter la palme" is used to emphasize the fact that the rent rates are high in "Les Grisons". It was probably in an article talking about the places where the rent rates are high.