You have to follow the steps as below:
1.Go to the Google Developers Console -https://console.developers.google.com
Select a project, or create a new one.
2. In the sidebar on the left, expand APIs & auth. Next, click APIs. In the list of APIs, make sure the status is ON for the YouTube Data.
Click the link ‘YouTube Data API’ from YouTube APIs section.
3. Then make sure that Enable API button is applied.
4. In the sidebar on the left, select Credentials.
5. Select the 'Create new Key from Public API' access. And choose Browser Key.
6. Create the Browser Key. And copy it to clipboard.
7.Paste the key to admin panel. For that you need to go to Advance Settings. and put "Last used Youtube API Key" in settings. Then click on Update Options to save it properly.
For detail information and screen shots please find the attached document.