Because they're two different l' You have to work it out from context. Also 'invito' is a noun here an invitation. If it was 'I invite him' then it would be Io lo invito or possible Io l'invito where it is the present tense conjugation of invitare (I invite) . But that's not the case here. If you wanted to say 'thank you for inviting him it might be Grazie per lo inviti. or Grazie per l'inviti or Grazie per l'invitate. But I don't know enough about the language to know if the verb would happen after the per like that.
But 'Thanks for I inviting him' doesn't make much sense.
At both normal and slow speeds I hear her say INvito. Both the Collins and Garzanti dictionaries give the pronunciation as inVIto. While any widely used language will have differences in pronunciation (sometimes as part of a dialect), would it not be best to give beginners the standard pronunciation?