Я знаю это правило наизусть.

Because правило is a neuter singular noun.

In Russian, words like this/that must agree with the noun in gender, number, and case:

• этот = masculine
• эта = feminine
• это = neuter
• эти = plural

Since правило is neuter, the correct form is это:

• это правило = this rule

Also, in context, это can sometimes be translated as this or that, depending on the situation.

Because правило is the direct object of знаю, so it is in the accusative case.

The verb знать usually takes a direct object in the accusative:

• Я знаю слово.
• Я знаю правило.

But here is the important part: for many inanimate neuter singular nouns, the accusative form is the same as the nominative form.

So:

• nominative: правило
• accusative: правило

That is why the form does not change, even though the case has changed.

Наизусть means by heart, from memory, or memorized perfectly.

So:

• Я знаю это правило наизусть. = I know this rule by heart.

It is often used when talking about something learned so well that you can say it without looking:

• знать наизусть = to know by heart
• выучить наизусть = to learn/memorize by heart
• читать наизусть = to recite from memory

Because in modern Russian it is a fixed adverb, and it is written as one word.

You should learn it as a whole expression:

• наизусть = by heart

It does not change form, and you do not usually analyze it as separate pieces in normal learning. The important thing is simply to recognize it as an adverb meaning from memory / by heart.

Because languages do not match word-for-word.

English says:

• know it by heart

Russian expresses the same idea with the adverb наизусть:

• знать наизусть

So the best approach is to learn the Russian expression as its own pattern, not as a literal translation of each English word.

Yes, absolutely.

• Я знаю это правило наизусть. = I know this rule by heart.
• Я знаю правило наизусть. = I know the/a rule by heart.

Including это points to a specific rule, like this rule. Without it, the sentence is more general or depends more on context.

Yes. In many contexts, Russian can drop the subject pronoun if it is already clear.

That is because the verb ending already shows who is doing the action:

• знаю = I know

So:

• Я знаю это правило наизусть. = full, neutral, explicit
• Знаю это правило наизусть. = natural in conversation when I is obvious

Russian uses subject pronouns less often than English does.

Because знать means to know in the sense of possessing knowledge.

Here, the idea is that the speaker has learned the rule so well that they know it from memory:

• Я знаю это правило наизусть.

This is more natural than using a verb like помнить here.

A useful contrast:

• знать наизусть = to know by heart
• выучить наизусть = to learn/memorize by heart

So if you are talking about the result, знаю is very natural.

Я знаю это правило наизусть is a very natural, neutral word order.

A simple breakdown:

• Я = subject
• знаю = verb
• это правило = object
• наизусть = adverbial expression

Russian word order is flexible, so you may also hear variations, especially for emphasis:

• Это правило я знаю наизусть.
  Emphasis on this rule
• Я это правило наизусть знаю.
  Less neutral; possible in speech for emphasis

But for learners, Я знаю это правило наизусть is the safest standard version.

The stress is:

• Я зна́ю э́то пра́вило наизу́сть.

A rough pronunciation guide:

• Я = ya
• зна́ю = ZNA-yu
• э́то = EH-tuh
• пра́вило = PRA-vi-luh
• наизу́сть = na-ee-ZOOST'

The most important stress points to remember are:

• зна́ю
• э́то
• пра́вило
• наизу́сть