It means “thank you“. In the standard Japanese, the pitch accent is あ↑りがとう (L-H-L-L-L) but in the Hiroshima dialect, it’s あり↑がとう (L-L-H-L-L).
この means “this“ which cannot be used by itself so it’s always followed by a noun.
And the following noun is 世界 meaning “world”.
The の particle is used to show position such as 机の上 meaning “on the desk”, テレビの横 meaning “next to the TV”.
片隅 means “a corner” so この世界の片隅 means “a corner of this world”.
And the に particle indicates the location so この世界の片隅に means “in this corner of the world” which is also used as the title of the movie.
うち means “I” or “me” in the Hiroshima dialect, used by females in casual speech. Men in Hiroshima use わし instead.
The を particle indicates the direct object of the following verb.
And the following verb is 見つけて which is the te-form of 見つける meaning “to find“.
So この世界の片隅にうちを見つける means “to find me in this corner of the world”, but you might wonder why the particle に is used instead of the で particle, as に indicates the location where something exists, whereas で indicates the location where an action takes place, and 見つける is an action.
Actually both に and で can be used in this case, but the nuance is different. Let’s compare:
- この世界の片隅にうちを見つける implies that you find me who exists in this corner of the world.
- この世界の片隅でうちを見つける implies that the action of finding takes place in this corner of the world.
So they both mean the same but the one with に focuses on existence whereas で is used to focus on action. I personally like に in this quote as it sounds like he managed to find her who was just in a corner of the world, which could have been difficult to find.
Lastly, くれて is the te-form of くれる meaning “to give me”. When it follows a verb te-form, it means “to do something for me“. And a verb te-form followed by くれてありがとう means “thank you for doing something for me”.
So it’s supposed to be この世界の片隅にうちを見つけてくれてありがとう meaning “thank you for finding me in this corner of the world”, but in this quote, the word order is inverted for emphasis.
Instead of ありがとう, you can also use 嬉しい, 幸せ and so on, to give positive feedback towards the action made by someone.