The best way you can help him at home is to help keep his poor speech from making him tense and anxious about talking. He seems to be telling you that he gets enough help at school and that he wants to be able to relax at home. Ask for an appointment with the person working with him at school and get suggestions for things you can do at home. I'm sure that the speech and language specialist won't want you to do exactly the same things at home that are being done at school. But I'm equally sure that the person will have recommendations for things you can do to help.
You sound like a very loving and concerned parent, so I am confident you will find the right way to help your son and the right level of involvement in his therapy. But, even though you desperately want to see his speech improve, you don't want to nullify gains he might be making at school by making him anxious and resentful of your efforts.
Our parenting advice is given as suggestions only. We recommend you also consult your healthcare provider, and urge you to contact them immediately if your question is urgent or about a medical condition.