To hide the post date and author on all pages—in archive post lists as well as on single posts—you can use some custom CSS…
article footer.entry-meta .date-meta,
article footer.entry-meta .author-meta { display: none }
Go to Appearance > Theme Options > Custom CSS and add that CSS to your Custom CSS box.
The post date displayed is the date the post was first made. If you want the date displayed to be the date a post was last modified you’d need to customise the theme’s ‘functions.php’ file. On line 339 of that file, or thereabouts, you’ll see this block of code…
// Date
$date = sprintf( '<span class="date-meta"><span class="meta-label">'. __( 'Date:', 'framework' ) .' </span><time class="entry-date" datetime="%1$s">%2$s</time>'.$sep.'</span>',
esc_attr( get_the_date( 'c' ) ),
esc_html( get_the_date() )
);
Change it to this…
// Date
$date = sprintf( '<span class="date-meta"><span class="meta-label">'. __( 'Date:', 'framework' ) .' </span><time class="entry-date" datetime="%1$s">%2$s</time>'.$sep.'</span>',
esc_attr( get_the_modified_date( 'c' ) ),
esc_html( get_the_modified_date() )
);
Save and upload.
Keep in mind that if you modify core theme files then each time you update the theme you will need to make all of those modifications again to the updated theme files.