Learn Dutch: Dit dialect is moeilijk te verstaan. - This dialect is difficult to understand.

This is an evaluative construction. When you combine an adjective like moeilijk with te plus an infinitive, the grammatical subject of the sentence (Dit dialect) is actually the receiver (or object) of the action. The people doing the understanding are left unspecified. English uses the exact same structure for sentences like this.

Could I use begrijpen instead of verstaan in this sentence?

You could, but it changes the nuance. Verstaan specifically refers to auditory understanding, meaning your ability to clearly hear and make out the sounds or words someone is saying. Begrijpen refers to intellectual understanding, like grasping a concept. Since dialects often involve unfamiliar pronunciation, verstaan is usually the perfect verb to describe the physical difficulty of making out the spoken words.

Verstaan has a prefix (ver-). Why doesn't te split the verb here, like it does in other sentences from this lesson?

The prefix ver- is inseparable, meaning verstaan is an inseparable verb. You only put te inside the verb if it is a separable verb, like uitspreken (to pronounce). If you wanted to say the dialect is hard to pronounce, the verb would split: Dit dialect is moeilijk uit te spreken.

Just a quick refresher: why is it Dit dialect and not ?