Apollo is indeed the God of Healing. Since he is the God of Light and Life, healing is associated with him. He is also using sun as a symbol together with bow.
Oedipus used Apollo's oracle which is located in his temple in Delphi to ask for Apollo's advice regarding unexplained plague which was occuring in Thebes that time. Phytia answered him. He asked his brother in law, Creon to see the plague. When he returns, he told Oedipus that it is a punishment from Gods. The Oracle also revealed a prediction that Oedipus will kill his father and will be married to his mother.
Apollo has an oracle because he uses it to communicate to people. He is using an Phytia, one of the priestess in the temple to send his message to everyone. His message can be composed of advices, future (predictions), questions and guidance to people, the Delphi city and other things.
The thing which Apollo has to do with the healing is that he is regarded as the god of healing so people call on him when they are sick or ailing.
The reason why Oedipus consults this god's oracle is that he wants to understand the plague which was destroying the city of Thebes.
The reason why Apollo has an oracle is that this is how he communicates with the people.
He is an Olympian deity in Greek and Roman cultures and has different powers and roles such as fertility, healing, archery, music, among others.
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