Learn Spanish (Spain): Ella no habla inglés y yo tampoco. - She doesn't speak English and neither do I.

In Spanish, languages and nationalities are not capitalized unless they are at the very beginning of a sentence. So, English is inglés and Spanish is español.

What exactly does tampoco mean here?

Tampoco is the negative version of también (too or also). It translates to neither or not either. When you agree with a negative statement, like someone not speaking English, you must use tampoco instead of también.

Why doesn't the second part of the sentence have a verb, like yo tampoco hablo?

You can leave the verb out because it is already understood from the first half of the sentence. Saying yo tampoco is a natural and common way to say me neither or neither do I, just like in English!

Do I have to include the pronoun Ella at the beginning?

You do not strictly have to, but it is very helpful here. Because the verb form habla can mean he speaks, she speaks, or formal you speak, adding Ella makes it perfectly clear who you are talking about. It also sets up a clear contrast with yo (I) later in the sentence.