Well, mi = I. If you're confused by its seeming absence in sentences like 'tha còta orm', it's actually still there, but combined with the preposition air (= on) into a so-called prepositional pronoun: air+mi -> orm (= on me).
One of the web based lessons explained this: 'am wanting' is something happening right now, 'want' is a usual or repetitive state. These two constructs are distinct in Scottish Gaelic, but not in standard English.
Yes - I think it's helpful for the understanding of this to think of "wanting" in the older (in English construction) sense of "lacking" not "desiring." So Tha mi ag iarraidh means something like"There is at me a lack of"