Learn Turkish: Onun çölde safari yapmışlığı var, şu an aslanları izliyordur. - He has the experience of having gone on a safari in the desert, he is probably watching the lions right now.

This is the experiential nominalization structure you are learning. The suffix -mış acts like a past participle, and -lık turns it into an abstract noun meaning 'the state of having done'. When we add the third-person possessive suffix -ı (causing the k to soften to ğ), we get yapmışlığı ('his state of having done'). Paired with the genitive pronoun onun and the word var, it literally means 'His state of having done a safari exists.'

Could I just say O çölde safari yaptı instead? What is the difference?

You could, but the focus shifts. O çölde safari yaptı is a simple past tense statement meaning 'He went on a safari in the desert.' Using Onun ... yapmışlığı var emphasizes the life experience itself, much like the English present perfect 'He has been on a safari' or 'He has the experience of doing a safari.'

Why does izliyordur have both the present continuous suffix -iyor and the suffix -dur?

When the suffix -dir / -dur is added to a present continuous verb (-iyor), it adds a sense of assumption, deduction, or strong probability. Instead of a factual 'he is watching', it translates to 'he is probably watching' or 'I assume he must be watching.'

Why does aslanları have the accusative suffix -ı here?