Learn Arabic (Modern Standard): ازداد الشاب نضجا بعد الرحلة. - The young man increased in maturity after the trip.

This is the Accusative of Specification, known in Arabic as التمييز. After verbs related to increasing, decreasing, or filling, like ازداد, Arabic uses an indefinite accusative noun to specify exactly what increased. Here, it tells us the young man increased specifically in maturity.

Does this specifying noun always have to be indefinite?

Yes, a تمييز must be indefinite. That is why we use نضجا instead of the definite version النضج. If you wanted to use the definite form, you would have to use a preposition instead, such as في النضج (in maturity).

The verb ازداد comes from the root ز ي د, so why is there an extra د in the middle instead of the usual Form VIII ت?

When a verb root starts with ز, the inserted ت of a Form VIII verb changes into a د to make it easier to pronounce. So instead of the awkward-sounding iztada, it becomes ازداد.

Could we use the Form I verb زاد instead of the Form VIII ازداد here?

Yes, زاد can also take the Accusative of Specification. However, ازداد is exclusively intransitive, meaning it only describes a subject that increases on its own. Because زاد can also be transitive (meaning to increase or add something else), using ازداد leaves no ambiguity that the young man himself is the one who grew.