Learn Polish: Jej oczy nagle zabłyszczały. - Her eyes suddenly sparkled.

The prefix za- adds an "inchoative" meaning, which means it marks the sudden start of an action or state. While błyszczały (imperfective) would mean her eyes "were sparkling" over a period of time, zabłyszczały (perfective) means they "flashed" or "started to shine" in a single, sudden transition.

Since zabłyszczały already implies a sudden start, isn't the word nagle (suddenly) redundant?

Technically, yes! Because the prefix za- already carries the idea of a sudden onset, the verb alone would be enough. However, adding nagle is very natural in Polish and often used for dramatic emphasis, making the suddenness of the reaction even more vivid.

Why does the past tense verb end in -ały instead of -eli?

The verb must agree with the subject, which is oczy (eyes). Even though they are the eyes of a woman, the noun oczy itself belongs to the non-masculine-personal (non-virile) category. Therefore, it takes the non-virile plural ending -y (resulting in zabłyszczały), rather than the masculine-personal -eli ending.

Is it strictly necessary to include jej (her) in this sentence?

No. If it is already clear from the context whose eyes you are talking about, you can easily drop jej. Polish frequently leaves out possessive pronouns when referring to body parts. Keeping it here just provides a slightly more deliberate, literary feel.