De docent legt het anders uit, zodat iedereen het beter begrijpt.

Because uitleggen is a separable verb.

• Whole verb: uitleggen = to explain
• In a main clause, the conjugated part goes to the usual verb position, and the prefix moves to the end:
  • De docent legt het uit.
• In an infinitive or many subordinate clauses, it stays together:
  • De docent wil het uitleggen.
  • ... omdat de docent het uitlegt.

So legt ... uit is exactly what you should expect in a normal main clause.

Dutch main clauses usually follow the V2 rule: the finite verb goes in the second position.

Here the first position is De docent, so the finite verb legt comes next:

• De docent | legt | het anders uit

This is one of the biggest differences from English word order.

In Dutch, short unstressed object pronouns like het often come before adverbs such as anders.

So:

• De docent legt het anders uit.

sounds natural.

A useful pattern is:

• verb + pronoun + adverb + separable prefix

So here:

• legt = finite verb
• het = object pronoun
• anders = adverb
• uit = separable prefix

Here anders means differently or in another way.

It describes how the teacher explains something:

• legt het anders uit = explains it differently

It does not mean else here. It is an adverb modifying the verb phrase.

The two instances of het do different jobs in different clauses, even though they refer to the same thing.

• De docent legt het anders uit
  Here het is the thing being explained.
• zodat iedereen het beter begrijpt
  Here het is the thing everyone understands better.

So both het words refer back to the same idea, topic, or explanation.

Dutch does this very naturally, just like English can say:

• The teacher explains it differently so that everyone understands it better.

Zodat means so that.

It introduces a clause of result or intended result/purpose:

• De docent legt het anders uit, zodat iedereen het beter begrijpt.

The idea is:

• the teacher changes the explanation
• so that everyone understands it better

So zodat connects the first action with its effect or intended outcome.

Because zodat introduces a subordinate clause, and in Dutch subordinate clauses the conjugated verb usually goes to the end.

Compare:

• Main clause: Iedereen begrijpt het beter.
• After zodat: zodat iedereen het beter begrijpt

That final verb position is very typical in Dutch subordinate clauses.

Because iedereen is grammatically singular in Dutch, even though it refers to many people.

So Dutch says:

• iedereen begrijpt not
• iedereen begrijpen

This is similar to English:

• everyone understands not
• everyone understand

Beter means better and is the comparative form of goed.

So:

• het goed begrijpen = understand it well
• het beter begrijpen = understand it better

In this sentence, the idea is that everyone understands it better than before or better than with the previous explanation. Dutch does not need to state that comparison explicitly.

Not in this exact sentence.

Because the first clause is a main clause, the separable verb must split:

• De docent legt het anders uit.

You would get uitlegt when the verb is in a subordinate clause, for example:

• ... omdat de docent het anders uitlegt.

So:

• main clause → legt ... uit
• subordinate clause → uitlegt