Mepham moved to the Premier League in January for £12m following his arrival from Brentford, where he made 43 Championship appearances after joining the club's academy from non-league side North Greenford United in 2012.
"I sort of felt like my time in football was done," he told Sky Sports. "As I would do, obviously, getting released from Chelsea.
"Going on trial at QPR, Watford and getting turned away - you question your own belief and where you want to be. At that time I didn't feel confident in football. I just felt like carrying on trying would be a waste of time.
"Luckily, I had a change of heart and went for that Brentford trial and I'd like to think it has paid off.
"When I joined North Greenford, in my head I knew that was it really for me. I had the opportunities and was turned away, so where do you go next? Little did I know it would take me here."