Villa were willing to do business with Manchester United over the Argentina goalkeeper, but the market closed with Martinez still a Villan.
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Emery said on Friday: "Everything happens with players, maybe staying, maybe they were close to leaving, maybe other players were close to others or to join us.
"It's normal. I understand it. Staying here is good news, I am so, so happy. He is the best goalkeeper in the world. His commitment to Aston Villa is massive.
"He is always training to get his best, respecting Aston Villa. Emi is happy and now he knows his challenge is with Aston Villa and his national team."