Android Unit Testing: How to make a class more testable?

I have been developing android applications but didn't write any unit tests. Recently I started to learn about it and try to use JUnit to test my android applications.

I found that most of the times I get bugs in API calls, But I still can't understand how to write unit tests for them(& how to make the original code testable).

Let me explain the following function:

I am running a function call setOffenceList(). There are multiple actions happening inside the function.

i) Load the RestClient and pass the URL.

ii) RestClient talk to the JSON api and get the response

ii) I grab the response inside the onSuccess(String response) function

iii) Parse the JSON data and store inside an array

iv) If success I will show data in a list view (else show an error message)

This is the code:

public class OffenceFrag extends Fragment {


    @Override
    public View onCreateView(LayoutInflater inflater, ViewGroup container, Bundle savedInstanceState) {
        View view = inflater.inflate(R.layout.frag_offence, container, false);
        //run API call
        setOffenceList();
        return view;
    }

    private void setOffenceList() {
        String url = Paths.SITE_URL ;
        RestClient.get(url, null, new AsyncHttpResponseHandler() {

            @Override
            public void onStart() {
                Toast.makeText(getActivity(), "Loading offences...", Toast.LENGTH_SHORT).show();
            }

            @Override
            public void onSuccess(String response) {

                //Parse JSON
                JSONArray jsonArray;
                try {

                    JSONObject jsonObj = new JSONObject(response);
                    if(jsonObj.getString("status").equalsIgnoreCase("Success")){

                        jsonArray = new JSONArray(jsonObj.getString("data"));
                        if(jsonArray.length() > 0){
                            for (int i = 0; i < jsonArray.length(); i++) {

                                JSONObject row = jsonArray.getJSONObject(i);
                                OffenceORM off = new OffenceORM();
                                off.setOffenceId(row.getString("offence_id"));
                                off.setPhoto(row.getString("photo"));
                                off.setSubmittedBy(row.getString("submitted_by"));
                                offenceList.add(off);
                            }
                        }   

                        //Success: Show the list view
                        setOffenceAdapter();
                        Toast.makeText(getActivity(), "Successfully Loaded", Toast.LENGTH_LONG).show();

                    } else {
                        //Failed: show error message
                        Toast.makeText(getActivity(), "There are no offences submitted under project", Toast.LENGTH_LONG).show();
                    }

                } catch (Exception e) {
                    Log.e("exception", e.getMessage());
                }
            }

            @Override
            public void onFailure(Throwable error, String content) {
                Log.e("failed", error.getMessage());
            }

            @Override
            public void onFinish() {

            }
        });
    }


}//end

I can't really understand how can I write a test function to something like the above code.

Can you please show me how to break down this code to testable pieces and write unit test functions to them?

Thanks a lot!