addEventListener, "change" and option selection

Another way to make your code understandable is :

let selector = document.getElementById("Selector"); 
let result = document.getElementById("result"); 

// when client clicked on select element 
selector.addEventListener("click", () => {
// if default value is changed
  selector.addEventListener("change", () => {
  // if value switched by client
    switch (selector.value) {
      case "add":
      //do somthing with  , "add" value
        result.innerHTML = selector.value;
        break;  // then take break
      case "remove":
      //do somthing with  , "remove" value
        result.innerHTML = selector.value;
        break; // then take break
    }
  });
});

#Selector{
margin: 1rem 0;
}

<label for"Selector">chose an option:</label>
<select id="Selector" name="Selector" >
      <option value="add">Add new item</option>
      <option value="remove">Remove existing item</option>
</select>

<div id="result">option selected : "The default value is first option"</div>

The problem is that you used the select option, this is where you went wrong. Select signifies that a textbox or textArea has a focus. What you need to do is use change. "Fires when a new choice is made in a select element", also used like blur when moving away from a textbox or textArea.

function start(){
      document.getElementById("activitySelector").addEventListener("change", addActivityItem, false);
      }

function addActivityItem(){
      //option is selected
      alert("yeah");
}

window.addEventListener("load", start, false);

You need a click listener which calls addActivityItem if less than 2 options exist:

var activities = document.getElementById("activitySelector");

activities.addEventListener("click", function() {
    var options = activities.querySelectorAll("option");
    var count = options.length;
    if(typeof(count) === "undefined" || count < 2)
    {
        addActivityItem();
    }
});

activities.addEventListener("change", function() {
    if(activities.value == "addNew")
    {
        addActivityItem();
    }
});

function addActivityItem() {
    // ... Code to add item here
}

A live demo is here on JSfiddle.