Learn Hardcore Greek: Φροντίζει την εμφάνισή του.

Greek often leaves out the subject pronoun because the verb ending already shows the person and number. Φροντίζει is 3rd person singular, so it means he/she/it takes care. The exact subject is understood from context.

What form is φροντίζει?

It is the present tense, 3rd person singular of φροντίζω. Here it means takes care of, looks after, or cares about. The present can describe a habit or general behavior, not only something happening right now.

Why is it την and not τη?

The feminine singular accusative article is often written as τη(ν). The final -ν is kept before vowels, and εμφάνιση starts with a vowel, so you write την εμφάνιση. That is why την is required here.

Why is there an article at all? English says his appearance, not the his appearance.

Greek normally uses the definite article with possessive expressions. So την εμφάνισή του is the natural Greek way to say his appearance. This is very common in Greek: το βιβλίο μου = my book, η αδερφή της = her sister.