The black border is showing because it’s a transparent png. To fix it, just assign background image color with white color.
#page-banner { background-color: #fff; }
But this kind of image (taken photo) is best saved in jpg, not png. The image now in png is 2.7MB which is way too much and it will slow the site down on every page load. Save it in jpg, with white background under it, there we will have the white border already in it.
Another issue is that the front page is missing the title, it’s better to have the header there for accessibility. So put in the page title, use something that’s useful to SEO like site title or business name and we can hide it with proper CSS.
/* ----- Front page title hide visually ----- */
/* --- https://github.com/h5bp/html5-boilerplate/blob/v5.0.0/src/css/main.css#L126 --- */
.home.page .hentry .entry-header {
border: 0;
clip: rect(0 0 0 0);
height: 1px;
margin: -1px;
overflow: hidden;
padding: 0;
position: absolute;
width: 1px;
}
/* ----- ml-slider plugin flexslider remove bottom margin ----- */
#page-banner .flexslider { margin: 0; border-bottom: 5px solid #adce80; }
/* ----- ml-slider plugin flexslider image background ----- */
#page-banner { background-color: #fff; }