Begin by setting up a Twilio account if you haven't already. Log in to your Twilio Console and navigate to the Dashboard. Here, you will find your Account SID and Auth Token, which are required for authentication when accessing Twilio's API. Keep these credentials secure as they provide access to your Twilio account.

Ensure you have Python installed on your local machine, as it will be used to interact with Twilio's API and manage data. You can download Python from [python.org](https://www.python.org/downloads/). Additionally, install the `twilio` Python package by running `pip install twilio` in your command prompt or terminal.

Create a new Python script file, e.g., `fetch_twilio_data.py`. This script will be used to connect to Twilio's API, fetch data, and save it to a local JSON file. Open the file in your preferred text editor or IDE.

In your Python script, import the necessary libraries and authenticate with Twilio's API. Use the following template:

```python
from twilio.rest import Client

# Your Account SID and Auth Token
account_sid = 'your_account_sid'
auth_token = 'your_auth_token'

# Create a Client instance
client = Client(account_sid, auth_token)
```

Replace `'your_account_sid'` and `'your_auth_token'` with your actual Twilio credentials.

Determine which data you need to extract (e.g., messages, calls, etc.). Use the Twilio Python SDK to fetch this data. Here’s an example for fetching SMS messages:

```python
messages = client.messages.list()

# Prepare data for JSON conversion
messages_data = []
for message in messages:
messages_data.append({
"from": message.from_,
"to": message.to,
"body": message.body,
"date_sent": str(message.date_sent)
})
```

This code snippet fetches SMS messages and stores relevant data in a list of dictionaries.

Use Python's built-in `json` module to convert the fetched data into JSON format:

```python
import json

# Convert the messages data to JSON format
json_data = json.dumps(messages_data, indent=4)
```

`json.dumps` serializes the Python list of dictionaries into a JSON-formatted string.

Finally, write the JSON data to a local file:

```python
# Write JSON data to a file
with open('twilio_data.json', 'w') as json_file:
json_file.write(json_data)

print("Data has been successfully saved to twilio_data.json")
```

This will create a file named `twilio_data.json` in your current directory and store the fetched data in JSON format.

By following these steps, you can effectively move data from Twilio to a local JSON file without relying on third-party connectors or integrations.