When I try to edit the page in SharePoint Designer it assk to checkout and edit the page layout, but I am not able edit the page in designer.
I want to edit List view web part code in page, but it not possible to edit page
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You're using a publishing page. A publishing page is made of a publishing layout and a master page.
The master page is responsible for displaying all the "chrome" (navigation, menus...) and the layout for the body area structure (user-authored publishing content).
In SharePoint Designer, you can edit both the master page or the layout. But everything else (including Web parts) is content added/edited by users Inside the placeholders of the content areas of the layout!
So, to edit Web parts on a publishing page, the only option is to switch the page to edit mode from the browser, and edit the Web part properties.
If you need super-tuning of a Web part, you can also super-tune it on a standard (non-publishing page) from SPD, export it as an WML file, and import it into the publishing page while in edit mode (from the Insert Web part panel).