The refusal of the engagement between Cecily and Algernon is done by Jack Worthing, who is Cecily's guardian. The reason why he does not want Algernon anywhere near Cecily is two-fold.
First, Jack is well aware of Algernon's bad behavior and irresponsible tendencies. Of all the men in the world, Algernon is really the least suited for an innocent, country estate heiress of which Jack has complete responsibility.
Second, Algernon had the nerve to visit Jack without invitation at his country estate expressly to meet and fall in love with Cecily, whom he had idealized. However, Cecily had also idealized Algernon, thinking that he was the fake "Ernest" from London: Jack's libertine brother. Under the fake persona of Ernest, Algernon is able to win over Cecily. However, Lady Bracknell's insistence in that Jack is not good enough for her own daughter, Gwendolen.
However, after seeing that Cecily came from money and that she would be a great way to get rid of Algernon and his libertine ways, Lady Bracknell declared that she would be more than happy to accept her in her household. This was a great chance for Jack to block the marriage, since Lady Bracknell did not let him marry Gwendolen. Hence, it was a good strategy to ensure that Lady Bracknell accepts the proposal by depriving her of the chance of getting rid of Algernon and marry him off to a wealthy country girl.