Learn Arabic (Modern Standard): بقيت في مصر ثلاث سنوات.

In Arabic, numbers 3 to 10 have reverse agreement with the noun they count. To figure out the gender, you always look at the singular form of the noun. The singular for years is سنة which is feminine. Because the counted noun is feminine, the number takes the masculine form ثلاث.

Why do we use the plural سنوات here, and what is its grammatical case?

For numbers 3 through 10, the counted noun must always be plural and in the genitive case. Instead of the singular سنة, we use the plural سنوات. If you were pronouncing the final vowels, it would end with a double kasra sound to show this genitive state.

The English translation says for three years, but there is no Arabic word for for in the sentence. Why?

In Arabic, you often do not need a preposition to express duration. While English requires a word like for, Arabic simply places the time phrase ثلاث سنوات directly after the verb to express how long the action lasted.

How does the word بقيت mean I stayed? Where is the pronoun?

The pronoun is built right into the verb. The base verb in the past tense is بقي. When talking about yourself, you add the suffix ـت to the end. This makes it بقيت, which means I stayed. You do not need to add a separate word for I.