。
It was around the time I'd started going to the pool for exercise.
Literal
Exercise [purpose-のため] pool [to-にも] going started time [copula-でした].
A temporal scene-setting sentence in narrative past. 運動のため ('for the purpose of exercise') uses ため (purpose/reason). ~にも通い始めた ('had also started going to') chains several elements: に (destination) + も ('also,' implying this was one of several new activities) + 通う (to commute/go regularly) + 始める (to start). 頃でした ('it was around the time when') uses 頃 (ころ, 'approximate time') with the copula in past polite form. Reads like the opening of a personal anecdote.