Après le cours de français, cette règle de grammaire commence à prendre du sens.

Après means after. The phrase Après le cours de français sets the time frame for the rest of the sentence: this is happening after French class.

French often puts this kind of time expression at the beginning of the sentence, just like English can do:

After French class, this grammar rule is starting to make sense.

You could also move it later in French, but the version here is very natural.

In French, de + subject/topic is very common for saying what something is about.

So:

• un cours de français = a French class / a French lesson
• un livre de grammaire = a grammar book
• un professeur de musique = a music teacher

If you say un cours français, that sounds more like a French course in the sense of a course that is French, not a course about the French language. So de français is the normal choice here.

When French names a language, it is usually written with a lowercase letter:

• le français
• l’anglais
• l’espagnol

So un cours de français is correct.

But when the word refers to a person’s nationality as a noun, it takes a capital letter:

• un Français = a Frenchman
• une Française = a Frenchwoman

As an adjective, it stays lowercase:

• un livre français
• une école française

Because règle is a feminine singular noun.

French demonstrative adjectives change depending on gender and number:

• ce = masculine singular before most consonants
• cet = masculine singular before a vowel or silent h
• cette = feminine singular
• ces = plural

So:

• ce livre
• cet exercice
• cette règle
• ces règles

Since règle is feminine, you need cette.

This is another very common noun + de + noun pattern in French.

Une règle de grammaire means a grammar rule. The second noun tells you what kind of rule it is.

Similar examples:

• une règle de sécurité = a safety rule
• un cours de maths = a math class
• un livre d’histoire = a history book

French often uses de where English uses one noun directly before another.

After commencer, French commonly uses à + infinitive when you want to say to start doing something.

So:

• commencer à comprendre = to start understanding
• commencer à parler = to start speaking
• commencer à prendre du sens = to start making sense

Here, prendre stays in the infinitive because it follows commence à.

You may see commencer de + infinitive, but for most learners, commencer à + infinitive is the safer and more natural choice in everyday modern French.

So:

• commence à prendre du sens = the best choice for normal use

Commencer de can sound more literary, formal, or old-fashioned depending on the context.

Prendre du sens is an idiomatic expression. It means something like:

• to start making sense
• to begin to seem meaningful
• to come together in your mind

It suggests a process: at first the rule was unclear, but now it is becoming clearer.

Compare:

• Cette règle a du sens. = This rule makes sense.
• Cette règle prend du sens. = This rule is starting to make sense / is becoming clearer.

So prendre du sens focuses on a change.

In this expression, du is the normal form.

Du is the contraction of de + le, but in expressions like avoir du sens or prendre du sens, you should think of it as part of the fixed phrase, not translate it word for word.

• avoir du sens = to make sense
• prendre du sens = to start making sense

Using un sens would usually change the meaning and suggest a meaning or one meaning, not the general idea of something making sense.

So here, du sens is simply the idiomatic form you should learn as a whole.

Not absolutely necessary, but it is very natural.

French often uses a comma after a longer introductory phrase, especially a time phrase, to make the sentence easier to read:

• Après le cours de français, cette règle de grammaire commence à prendre du sens.

Without the comma, the sentence is still understandable:

• Après le cours de français cette règle de grammaire commence à prendre du sens.

But the comma helps separate the introduction from the main clause, so it is a good stylistic choice.

Because the subject is cette règle de grammaire, and the main noun in that subject is règle, which is singular.

Even though the subject contains several words, the verb agrees with the head noun:

• Cette règle de grammaire commence...
• Ces règles de grammaire commencent...

So commence is singular because règle is singular.

Yes, grammatically you could, but commence à prendre du sens sounds more natural and idiomatic here.

There is a small nuance:

• avoir du sens = to make sense
• prendre du sens = to become meaningful / to start making sense

Since the sentence describes something gradually becoming clear after class, prendre du sens fits especially well.