Uploading media with the REST API

This is not possible via the API; you need to fetch the image yourself, and send the data to the API yourself.
Blockquote

^{– This quotes Ryan on the GitHub–issue mentioned in @Dan (deleted) answer.}

How to side load images

Note that the media_sideload_image() return value can be an instance of \WP_Error as well. You then just need to attach it to a post, which is quite simple:

$media = media_sideload_image( 
    "http://i.imgur.com/bcJvAj0.jpg', 
    $post->ID, 
    'Some image description', 
    'src'
);

if ( ! empty( $media ) and ! is_wp_error( $media ) ) {
    // reference new image to set as featured
    $attachments = get_posts( [
        'post_type'      => 'attachment',
        'posts_per_page' => 1,
        'post_status'    => 'any',
        'post_parent'    => $post->ID,
    ] );

    if ( is_array( $attachments ) ) {
        set_post_thumbnail( $post->ID, $attachments[0]->ID );
    }

    // Test print our image. The return value is the src of the sideloaded image.
    printf( '<img src="https://wordpress.stackexchange.com/questions/229430/%s" />', $media );
}

The internals

A short back trace through core that explains why you should stick with the API:

Take a look at the source of media_sideload_image() to see it uses download_url() internally, which uses wp_safe_remote_get(), a wrapper for \WP_HTTP::get(). This means it offers the whole WP API while being quite safe, debuggable and easy to use. The return value of download_url() is the result of wp_tempnam(), which takes all possible /tmp locations into account (there are plenty) and makes sure your location is writeable. The media_handle_sideload() will generate all needed attachment data and meta data.