To open a browser window inside VS Code you can use the WebView API, though you need to supply HTML content rather than a URL:
export function activate(context: vscode.ExtensionContext) {
  context.subscriptions.push(
    vscode.commands.registerCommand('catCoding.start', () => {
      // Create and show panel
      const panel = vscode.window.createWebviewPanel(
        'catCoding',
        'Cat Coding',
        vscode.ViewColumn.One,
        {}
      );
      // And set its HTML content
      panel.webview.html = getWebviewContent();
    })
  );
}
function getWebviewContent() {
  return `<!DOCTYPE html>
<html lang="en">
<head>
    <meta charset="UTF-8">
    <meta name="viewport" content="width=device-width, initial-scale=1.0">
    <title>Cat Coding</title>
</head>
<body>
    <img src="https://media.giphy.com/media/JIX9t2j0ZTN9S/giphy.gif" width="300" />
</body>
</html>`;
}

Depending on your specific use case, there's also the Browser Preview extension which registers a command browser-preview.openPreview that you could use (you'd probably want to list browser-preview as a dependency to ensure it's installed).
And finally, if you just want to open in a normal browser window you can use the env.openExternal API. When running in remote environments this will also take care of exposing ports and mapping to the exposed hostname (if it's a localhost-served service):
vscode.env.openExternal(Uri.parse("https://www.stackoverflow.com/"));