Мой день быстро проходит, когда я работаю.

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Breakdown of Мой день быстро проходит, когда я работаю.

• мой: my
• день: the day
• когда: when

Questions & Answers about Мой день быстро проходит, когда я работаю.

Why is проходит used here instead of идёт or идет (as in день идёт)? Don’t they both mean “the day goes by”?

Both проходит and идёт can be used about time, but they have slightly different nuances.

• день проходит – the day passes, goes by, often with a sense of its duration being used up, “the day is passing (and eventually ends).”
• день идёт – literally “the day is going,” more neutral, like “time is moving forward.”

In this sentence, Мой день быстро проходит emphasizes that your day is passing quickly, that it is over before you know it.
You could say Мой день быстро идёт, когда я работаю, and it would be understandable, but проходит is more idiomatic here when talking about how fast the day is passing.

What is the grammatical form of проходит, and what does it tell me?

Проходит is:

• verb: проходить
• aspect: imperfective
• tense: present
• person/number: 3rd person singular (он/она/оно проходит)

So день проходит = the day passes / is passing (in general, regularly, or now).
Present tense + imperfective aspect here describes a regular, repeated situation: every time you work, your day tends to pass quickly.