To integrate an external API into a WordPress page, you can use either a theme modification or plugin integration approach. Theme modifications involve creating a child theme and adding a custom page template to interact with the API. This process involves five steps: getting an API key for the REST API, creating a child theme, creating a custom page template, integrating the API into your page template, and adding a new page using your new page template. The example provided uses the Next Generation Netflix Global Search API to display up to 100 of the most recent items added to Netflix within the last 7 days, with the results displayed in a grid format on the custom page template. Once the child theme is activated, you can choose this custom page template when adding a new page in the WordPress admin area, and apply basic CSS to style the output as desired.