So break this problem down into pieces. First you want to grab the 'username' from your $SESSION variable, right?
session_start();
if (isset($_SESSION['username'])) {
$user = $_SESSION['username'];
} else {
echo "<h1>No user is logged in!</h1>";
}
Once you have that, put it into an SQL query:
$sql = "SELECT day, month, year FROM Booking WHERE username = '".$user."'";
$result = mysql_query($sql, $link)
or die("<p>Couldn't execute query: <strong>".mysql_error."</strong>. Problem with this query: <strong>".$sql."</strong>");
while ($row = mysql_fetch_array($result)) {
echo "<li />Dates are: ".$row['day']." ".$row['month']." ".$row['year'];
}