The 53-year-old's future at the club had been uncertain for some time, with owner Massimo Cellino actively searching for a replacement while Evans was still in the role.
"First and foremost I have always believed that, if you're an owner and have the final say, you're duty bound to put the best man in the chair for the job," he told Sky Sports News.
"If Mr Cellino has an option on me to extend my contract and sees fit to speak to other people, I had to ask myself, 'One, was he doing it because he doubted me or, two, he wanted to see if there was something better in the market?' For a time I believed it was the second of those two, but the last week I saw there were other people in for the job and I took the opinion it was not going to be me.
"It was hard to sit there and accept. When you work for Leeds United, the first thing you realise is the size and the fans are simply stunning and wonderful people and you realise you have to keep your dignity and respect. If you come out of that club it's a sad day. I will move on, but it's been a wonderful experience."
However, despite Cellino's public search for a new manager before Evans had left the club, the former Rotherham United boss insists that he did not feel undermined by the controversial owner.
"No, not at all. I wish that great football club and those supporters every success and I will be the first to buy a ticket if they get in the Premier League," he added.
Cellino has now sacked six coaches since arriving at the club two years ago.