I thought I'd post this here in case it may help someone else. It took a while to figure out.
Backdrop's Form API provides the element managed_file, which allows you to add a widget to your forms to upload new managed files. The documentation for this element is good, except that it doesn't contain much about the property #browser_view, other than saying that it allows developers to indicate a View to use in the file browser dialog.
So, my use case was: I needed to provide a file browser that allowed the user to pick PDF documents from the managed files (not images). These are the steps:
1. Create your form and include the managed_file element. For example:
$form['certificate_template_file'] = array(
'#type' => 'managed_file',
'#title' => t('Template PDF file'),
'#description' => t('The template PDF file to be used for this certificate.'),
'#default_value' => $certificate_template->certificate_template_file,
'#upload_location' => 'public://certificate_templates',
'#upload_validators' => array(
'file_validate_extensions' => array('pdf'),
'file_validate_size' => array(5 * 1024 * 1024),
),
'#browser_view' => 'pdf_doc_library',
);
The above indicates that the View "pdf_doc_library" will be used for the file browser.
2. Create your View. Show: Files. I used a Grid display.

3. This is important: in the Grid settings, add the Row class "file-browser-file". Backdrop's file browser will look for that class to select the file ID when clicked.
4. Notice the hidden fields Path and Name (image above). The Path field MUST have the option "Use full URL instead of file storage URI" clicked.
The other important part: rewrite the File ID fields as follows:

The important part here is to provide a data-file-url, AND a data-fid with the data from the hidden fields. The rest you can customize as you wish.
5. Then a bit of styling added to a CSS file that gets loaded with the form, such as:
.file-browser-file:hover {
background-color: #e5f3ff;
}
.file-browser-selected {
background-color: #cce8ff;
}
That's it. Then, in the form, when clicking "Select existing files" you'll get:

A minor annoyance: the dialog has the title "Select from Image Library". That title is, unfortunately, hard-coded in file_managed_file_browser_open(). Perhaps I'll create an issue for this. It'd be easy to pass the title within $form_state.
Recent comments
well @herb I cannot believe it, but the webform module was there in modules on the disc, but missing in the listed modules, so I delete the module folder, and downloaded it again. Then...
My Webform Worksheets have disappeared
The only thing I can think of is to make sure Webform module is enabled and that the content type that needs to contain the webforms has Webform enabled for it. Edit the content type and make...
My Webform Worksheets have disappeared
@yorkshirepudding, In the end I fixed it by deleting the config/active folder contents, copying the config/staging/ files into config/active, importing the database again and clearing the caches...
Importing the staged config throws an error: another field with the same name already exists
Thanks, I have to go out now, I will try later on and report back
Importing the staged config throws an error: another field with the same name already exists
Hi Francis. Yes, what config do you have in staging for it? You might have configs like: field.field.field_video_embed field.instance.node.page....
Importing the staged config throws an error: another field with the same name already exists