Danny Whitaker scored in the 94th minute to cancel out Cheltenham debutant Frankie Artus' 31st-minute strike and deny the relegation-threatened home side a third win in four games.
Royle said: "In the first half we were awful and I had a lot of words to say at half-time.
"I felt for the fans who had travelled down and were sitting in the freezing cold as we were weak and insipid.
"I was disillusioned at half-time and our performance had left me very down and very angry.
"But I was heartened at the way we played in the second half. We played some excellent football and carved out some good chances."