Another way to think of it is that the subject of a verb in the infinitive is in the dative. So the subject of отдыхать is Антону. There is a somewhat similar construction in English in sentences like "It is difficult for me to do" - in Russian, the "for me" would be replaced by the dative (though these constructions are much more common in Russian than in English - the хочется construction in English would sound like "It is desired for me to..." which we don't say).

In Russian it's even possible to have a sentence like "мне идти." with just the dative and the verb (it's one of the ways of saying "I need to go").