Learn Arabic (Modern Standard): الضيف شكرني.

It is the ـي attached to the very end of شكرني. In Arabic, pronoun objects like "me", "him", or "us" are never separate words. They always attach directly to the end of the verb.

What is the letter ن (noon) doing in the middle of شكرني?

The verb is شكر (he thanked) and the pronoun for "me" is ـي. If we joined them directly, it would force the verb to end with a sharp "ee" sound. Arabic verbs do not like this sound! To protect the verb's pronunciation, a buffer letter ن (called the "noon of protection") is inserted. That is why it is شكرني (shakara-ni) and not "shakari".

Since شكرني means "thanked me", how do I know who did the thanking?

The base verb here is شكر (shakara), which is the default "he" form in the past tense. Even if we completely removed the word الضيف (the guest), the word شكرني on its own would still mean "he thanked me".

Could I put the verb first and say شكرني الضيف?

Yes! Arabic allows both word orders. Starting with the noun الضيف makes it a nominal sentence, which puts slightly more focus on the guest. Starting with the verb makes it a verbal sentence. Both are perfectly correct and common.