La dentiste dit que j'ai une carie.

Because the subject is la dentiste, which is third-person singular: she says. The verb is dire and its present-tense forms include:

• je dis = I say
• tu dis = you say
• il/elle dit = he/she says

So la dentiste dit means the dentist says.

Que means that and introduces the second part of the sentence: que j'ai une carie = that I have a cavity.

In English, that is often optional: The dentist says (that) I have a cavity.
In French, que is normally required in this kind of sentence.

French usually shortens je to j' before a vowel sound. This is called elision.

So:

• je ai becomes j'ai

This is very common:

• j'aime
• j'habite
• j'ai

Ai is the je form of the verb avoir in the present tense.

So:

• j'ai = I have

In this sentence, j'ai une carie literally means I have a cavity.

In French, singular countable nouns usually need an article. Since carie is a feminine noun, the indefinite article is une.

So:

• une carie = a cavity

Saying just carie by itself would usually sound incomplete.

Noun gender in French is grammatical, not logical. Carie happens to be a feminine noun, so it uses feminine articles and adjectives:

• une carie
• la carie

This is something you generally have to learn with the noun.

It can mean both, depending on context. French present tense often covers both the simple present and the present progressive.

So La dentiste dit... can mean:

• The dentist says...
• The dentist is saying...

In most normal translations here, The dentist says that I have a cavity is the most natural choice.