Learn Russian: Тренируясь каждый день, спортсмен замечает свой прогресс. - Training every day, the athlete notices his progress.

It is a verbal adverb (or gerund) meaning 'while training.' It is formed from the imperfective verb тренироваться. You use this form to describe a secondary action happening at the same time as the main verb, as long as both actions are done by the same subject (the athlete).

Why does Тренируясь end in -сь instead of -ся?

The base verb тренироваться is reflexive. When making a verbal adverb from a reflexive verb, the reflexive particle shrinks from -ся to -сь after a vowel. So, instead of тренируяся, it becomes тренируясь.

Why use свой instead of его for 'his progress'?

The word свой is a reflexive possessive pronoun that refers back to the subject of the sentence. Because the athlete is noticing his own progress, you must use свой. If you used его прогресс, it would mean he was noticing some other man's progress.

Why isn't there a preposition before каждый день?

In Russian, phrases describing frequency or duration, like 'every day' (каждый день) or 'all year' (весь год), are put directly into the accusative case without needing a preposition.