Display Authors Comments on Profile Page

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What you need to use here is the WP_Comment_Query() function.

So on the author.php page, you can easily get the author info and ID as followed:

// get author info
$curauth = (isset($_GET['author_name'])) ? get_user_by('slug', $author_name) : get_userdata(intval($author));

// set ID
$user_id = $curauth->ID;

Then we add the user ID in the query arguments array:

$args = array(
    'user_id' => $user_id, // comments by this user only
    'status' => 'approve',
    'post_status' => 'publish',
    'post_type' => 'post'
);

And finally we hit the arguments in the wp_comment_query():

// The Query
$comments_query = new WP_Comment_Query;
$comments = $comments_query->query( $args );

// Comment Loop
if ( $comments ) {
    foreach ( $comments as $comment ) {
        echo '<p>' . $comment->comment_content . '</p>';
    }
} else {
    echo 'No comments found.';
}

As an added bonus, I investigated how pagination works with wp_comment_query() for not long ago and offer a good solution here. It was a little bit of a fiddle-niddle to make it work.

EDIT:

A better way to get the author ID is simply with (props @Pieter):

$user_id = get_queried_object_id();

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