Issues : You have not changed  your background colors on hover and active states. See
.our-work-side-menu ul li:hover {
    background-color: #529883; /*Same Color */
}
.our-work-side-menu ul li a {
    color: #529883;
}
.our-work-side-menu ul li a:hover { /* Repeated | Redundant CSS */
    color: #ffffff;
    background-color: #529883; /*Same Color */
}
.our-work-side-menu a:active {
    color: #ffffff;
    background-color: #529883; /*Same Color */
}
Fixes:
.et_pb_sidebar_0.et_pb_widget_area {
  border: 1px solid #a9a9a9;
}
.et_pb_sidebar_0.et_pb_widget_area .widgettitle {
  padding-left: 30px;
}
.our-work-side-menu ul li {
  box-sizing: border-box;
  margin: 0;
  border-top: 1px solid #a9a9a9;
  width: 100%;
}
.our-work-side-menu ul li a {
  color: #529883;
  display: block;
  height: 100%;
  text-decoration: none;
  padding: 10px 0 10px 40px;
}
.our-work-side-menu ul li a:hover {
  color: #ffffff;
  background-color: #529883;
}
.our-work-side-menu ul li a:active {
  color: #ffffff;
  background-color: orange;
}
 /* UPDATE: For clicked link behaviour we set an active class  */
 .active {
   background: orange;
   color: #ffffff !important;
 }
UPDATE :
JAVASCRIPT:
For Clicked Link to have the active style, we need to use a bit of CSS. I have come up with this code (though) there are many approaches :)
document.addEventListener("DOMContentLoaded", () => {
  let linksWrapper = document.querySelector(".links-wrapper");
  let hiddenLI = document.createElement("li");
  let anchor = document.createElement("a");
  anchor.setAttribute("class", "active");
  hiddenLI.appendChild(anchor);
  linksWrapper.appendChild(hiddenLI);
  hiddenLI.style.display = "none";
  const mainLinks = document.querySelectorAll(".links-wrapper li a");
  for (var i = 0; i < mainLinks.length; i++) {
    mainLinks[i].addEventListener("click", function () {
      var current = document.getElementsByClassName("active");
      current[0].className = current[0].className.replace("active", "");
      this.className = "active";
    });
  }
});
FULL WORKING CODE:
document.addEventListener("DOMContentLoaded", () => {
  let linksWrapper = document.querySelector(".links-wrapper");
  let hiddenLI = document.createElement("li");
  let anchor = document.createElement("a");
  anchor.setAttribute("class", "active");
  hiddenLI.appendChild(anchor);
  linksWrapper.appendChild(hiddenLI);
  hiddenLI.style.display = "none";
  const mainLinks = document.querySelectorAll(".links-wrapper li a");
  for (var i = 0; i < mainLinks.length; i++) {
    mainLinks[i].addEventListener("click", function () {
      var current = document.getElementsByClassName("active");
      current[0].className = current[0].className.replace("active", "");
      this.className = "active";
    });
  }
});
.et_pb_sidebar_0.et_pb_widget_area {
  border: 1px solid #a9a9a9;
}
.et_pb_sidebar_0.et_pb_widget_area .widgettitle {
  padding-left: 30px;
}
.our-work-side-menu ul li {
  list-style: none;
  box-sizing: border-box;
  margin: 0;
  border-top: 1px solid #a9a9a9;
  width: 100%;
}
.our-work-side-menu ul li a {
  color: #529883;
  display: block;
  height: 100%;
  text-decoration: none;
  padding: 10px 0 10px 40px;
}
.our-work-side-menu ul li a:hover {
  color: #ffffff;
  background-color: #529883;
}
.our-work-side-menu ul li a:active {
  color: #ffffff;
  background-color: orange;
}
.active {
  background: orange;
}
<!DOCTYPE html>
<html>
  <head>
    <meta charset="utf-8" />
    <meta http-equiv="X-UA-Compatible" content="IE=edge" />
    <meta name="viewport" content="width=device-width, initial-scale=1" />
    <link rel="stylesheet" href="style.css" />
  </head>
  <body>
    <div class="our-work-side-menu">
      <ul class="links-wrapper">
        <li><a href="#">Completed Projects</a></li>
        <li><a href="#">Ongoing Projects</a></li>
        <li><a href="#">Future Projects</a></li>
        <li><a href="#">Photo Gallery</a></li>
        <li><a href="#">Video Center</a></li>
        <li>
          <a href="#">Dr. Asamoah Bio Video</a>
        </li>
      </ul>
    </div>
  </body>
  <script src="./script.js"></script>
</html>