You can use JSON.parse() to convert the String into an object with objects = JSON.parse(events);
after that, access is possible to access it with objects[0].title
See further details about json.parse here: https://www.w3schools.com/js/js_json_parse.asp
Your code should look like
var events='[{"title":"Appointment","start":"10:30:00"}, {"title":"Appointment","start":"11:00:00"},{"title":"Appointment","start":"11:15:00"},{"title":"Appointment","start":"11:45:00"}]';
var objects = JSON.parse(events);
console.log(objects[0].title);
Another (but bad way) is using eval(). parsing the string with eval will execute the code and returns a result (In this case a javascript object) that can be accessed the same way. However, if you have malicous code in in the string, it will be executed as well. (See https://developer.mozilla.org/de/docs/Web/JavaScript/Reference/Global_Objects/eval)
If you can, use JSON.parse().
JSON.parse.