From Wiktionary, the free dictionary
Literally, “to get (something) into one’s hungry stomach”.
fá sér í svanginn
- (idiomatic) to get (something) to eat [with dative]
Það er alltaf gott að fá sér eitthvað í svanginn eftir langa göngu.- It always feels good to get something to eat after a long hike.