Ich lasse das Wasser erst aufkochen und gieße dann den Reis ab, wenn er fertig ist.

Yes, but in this kind of sentence lassen is often best understood as have or make in English, not only let.

Ich lasse das Wasser aufkochen means something like:

• I let the water come to a boil
• I bring the water to a boil

The pattern is:

lassen + object + infinitive

So here:

• ich = the person causing the action
• das Wasser = the thing affected
• aufkochen = the action that happens

This is a very common German structure.

aufkochen means to come to a boil, to boil up, or to bring something to the boil depending on context.

It is not exactly the same as kochen:

• kochen = to cook / to boil
• aufkochen = to boil up, to reach boiling point

So das Wasser aufkochen lassen focuses on the water reaching the boil.

Both aufkochen and abgießen are separable verbs.

In German:

• when the verb is in the infinitive, the prefix stays attached:
  • aufkochen
  • abgießen
• when the verb is conjugated in a main clause, the prefix separates:
  • Ich gieße den Reis ab.
  • Das Wasser kocht auf.

So in your sentence:

• lasse ... aufkochen → aufkochen is an infinitive, so it stays together
• gieße ... ab → gieße is a finite verb, so ab goes to the end

erst ... dann is a very common way to show sequence:

• erst = first
• dann = then

So the sentence is organizing the actions step by step:

• first: das Wasser aufkochen
• then: den Reis abgießen

Be careful: erst can also mean only / not until in other contexts, but here it clearly means first.

Because den Reis is the accusative form.

der Reis is masculine, so:

• nominative: der Reis
• accusative: den Reis

Here den Reis is the direct object of abgießen:

• Ich gieße den Reis ab.

Even though English often says drain the rice or pour off the water, German commonly uses the food itself as the object: den Reis abgießen.

abgießen means to pour off or, in cooking, to drain.

So den Reis abgießen means to drain the rice.

Literally, the idea is that you pour away the liquid. In kitchen German, this verb is used a lot for pasta, potatoes, rice, and similar foods.

er refers to der Reis.

German pronouns usually match the grammatical gender of the noun:

• der Reis → er
• das Wasser → es

So even though English would say when it is ready, German says wenn er fertig ist because Reis is masculine.

Because wenn is the normal word for when in this kind of time clause.

A quick comparison:

• wenn = when, whenever, if
  used for repeated situations, general truths, or future-related situations
• als = when
  used for a single event in the past
• wann = when
  used in questions or indirect questions

Here the sentence describes a general procedure or routine, so wenn is the right choice:

• ..., wenn er fertig ist. = ..., when it is done.

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

So:

• main clause: Er ist fertig.
• subordinate clause: wenn er fertig ist

That is normal German word order after words like:

• wenn
• weil
• dass
• obwohl

Because German normally puts a comma before a subordinate clause.

Since wenn er fertig ist is a subordinate clause, it must be separated with a comma:

• ..., wenn er fertig ist.

German comma rules are stricter than English here.

German uses the present tense very often for:

• routines
• instructions
• planned actions
• near-future meaning

So Ich lasse ... und gieße ... ab can sound perfectly natural even if the meaning is like:

• I first bring the water to a boil and then drain the rice when it’s done
• or even I’ll first bring... and then drain...

German does not need a future form here.

fertig here means ready or done.

In cooking, German often uses fertig by itself when the meaning is obvious. So:

• wenn er fertig ist = when it is done / ready
• more explicit: wenn er fertig gekocht ist = when it is fully cooked

So nothing is really missing; the shorter version is completely normal.