strtotime() is returning bool(false), because it can't parse the string '+one year' (it doesn't understand "one"). false is then being implicitly cast to the integer timestamp 0. It's a good idea to verify strtotime()'s output isn't bool(false) before you go shoving it in other functions.
There is also a simpler and less sophisticated solution:
$monthDay = date('m/d');
$year = date('Y')+1;
$oneYearFuture = "".$monthDay."/".$year."";
echo"The date one year in the future is: ".$oneYearFuture."";
// Declare a variable for this year
$this_year = date("Y");
// Add 1 to the variable
$next_year = $this_year + 1;
$year_after = $this_year + 2;
// Check your code
echo "This year is ";
echo $this_year;
echo "<br />";
echo "Next year is ";
echo $next_year;
echo "<br />";
echo "The year after that is ";
echo $year_after;