Learn Hardcore Japanese: sono isya ha eigo to nihongo no hutatu no gengo wo hanasimasu. - That doctor speaks two languages, English and Japanese.

その is the demonstrative adjective “that.” It refers to someone already known or mentioned in the conversation. By saying その医者, you mean “that doctor (we’ve been talking about).”

What does the particle は do after 医者?

は marks 医者 as the topic of the sentence. It tells the listener, “As for that doctor …,” and then the rest of the sentence comments on that topic.

Why is と placed between 英語 and 日本語, instead of や?

と joins nouns in a complete list (“English and Japanese”). If you used や, it would imply an incomplete list (“English, Japanese, and other languages, etc.”). Here you want to specify exactly those two.

Why is there a の after 日本語 in 英語と日本語の二つの言語?

That の connects the list of languages to the counter phrase 二つ. Literally it says “the two languages of English and Japanese,” with の showing possession/attribution.

Why do we say 二つの言語 instead of something like 二言語?

二つ is the general counter for “two (things).”
二言語 is more technical and less common in everyday speech.
Using 二つの言語 is natural and clear: “two languages.”

Could we drop 二つの言語 and simply say 英語と日本語を話します?

Yes.

その医者は英語と日本語を話します。 already means “That doctor speaks English and Japanese.”
Adding 二つの言語 makes the number explicit or stylistically more formal.

Why is the object marker を used before 話します?

話します is a transitive verb meaning “to speak (something).” The things being spoken—英語と日本語—are its direct object, so they take を.

What’s the difference between 話します and 話せます?

話します (hanashimasu): “speaks” or “will speak.” It describes the action of speaking.
話せます (hanasemasu): “can speak.” It’s the potential form, expressing ability.
To say “That doctor can speak English and Japanese,” you’d say その医者は英語と日本語が話せます。