Learn Spanish (Latin America): El caballo se echa a galopar y el jinete mira el cronómetro. - The horse starts to gallop and the jockey looks at the stopwatch.

Echar(se) a + infinitive is a sudden inchoative periphrasis, meaning it emphasizes that an action starts abruptly or unexpectedly. While empieza a just means "starts to," se echa a galopar gives the feeling that the horse suddenly burst into a gallop.

I've seen echar a used without the se. Why is it se echa a here?

Both echar a and echarse a mean to suddenly start an action. Traditionally, non-reflexive echar a is used for verbs of motion, while echarse a is for emotional reactions (like llorar). However, in Latin American Spanish, it is extremely common to use the reflexive echarse a with verbs of motion as well, which is why you see se echa a galopar here.

Can I use romper a here instead of echarse a, since the lesson covers both?

No, romper a wouldn't sound natural here. Romper a is also used for sudden beginnings, but it is almost exclusively paired with a few specific expressions of emotion or weather, like romper a llorar (to burst into tears) or romper a llover (to start pouring rain). For physical motion like galloping, echar(se) a is the right choice.

Why doesn't the sentence say mira al cronómetro since it translates to "looks at the stopwatch"?