Learn Turkish: Sen çok meşgulsün, bu işi ben halledeyim. - You are very busy, let me sort out this work.

It comes from the verb halletmek (to handle/sort out). To make the "let me" form (optative mood) taught in this lesson, we add -eyim to the root hallet. Because the root ends in t and is followed by a vowel, the t softens to a d, giving us halledeyim.

Wait, doesn't vowel harmony mean it should be meşgulsun instead of meşgulsün?

Normally, yes! However, meşgul (busy) is a borrowed word that ends with a "soft" L sound. In Turkish, this soft L makes the word behave as if its last vowel was a front vowel like ü. That is why it takes the "you are" ending -sün instead of the expected -sun.

Why is it bu işi instead of just bu iş?

Bu iş (this work) is the specific thing receiving the action of the verb (the thing being sorted out). Therefore, it needs the specific direct object ending (the accusative case). The suffix -i is added to match the vowel in iş.

Do we need to include ben before the verb since halledeyim already means "let me"?

Grammatically you can drop it, but it's used here for emphasis and contrast. The sentence sets up a clear shift in responsibility: "You (Sen) are very busy, so I (ben) will handle this work."