Glad to see so many people thought it was "my clothes" too. It's always helpful at the start of a lesson to be reminded that we should calm down the English speaker in our brains and remember that just because a word sounds like "my", this is a different language where "my" is "meg" and "meg" is "me"!
"Lette" is used for all plurals. It's also used for definite singulars of all genders when attributively placed, i.e. placed in front of the noun.
"Lett" is used for indefinite singulars of all genders, and when predicatively placed (after the noun) modifying definite singulars:
"En lett bil" (indefinite singular)
"Bilen er lett" (definite singular; predicative placement)
"Den lette bilen" (definite singular; attributive placement)
"Lette biler" (indefinite plural)
"Bilene er lette" (definite plural; predicative placement)
"De lette bilene" (definite plural; attributive placement)
"Å ha på seg" is a reflexive verb in Norwegian, translating literally to "to have on oneself". So "meg" relates to the verb, not to the noun "klær".
The reflexive pronoun agrees with the subject, so you get "jeg har på meg", but "du har på deg", "vi har på oss", and so on.
Other comments indicate that the purpose of "meg" in this sentence is to make the verb reflect that it is first person singular. But I have not seen that practice with any other verbs. My aim with this comment is to understand why this sentence needs "meg" to perform this function, whether other verbs exhibit similar behaviors, and how to know which (if any) do and how to use them properly. A thousand thanks in advance!