You can get the display name of users from people picker field using below code:
var peoplePickerID = "TestPersonField";
var clientPeoplePicker = SPClientPeoplePicker.SPClientPeoplePickerDict[$('[title^="' + peoplePickerID + '"]')[0].id];
var peoplePickerUsersInfo = clientPeoplePicker.GetAllUserInfo();
if (peoplePickerUsersInfo.length > 0) {
for(var index = 0; index < peoplePickerUsersInfo.length; index++) {
console.log(peoplePickerUsersInfo[index].DisplayText);
}
}
Output:

Update from comments:
If above code is not working for you, you can use below code to extract "DisplayName" from "ProcessedUserList":
var peoplePickerID = "TestPersonField";
var clientPeoplePicker = SPClientPeoplePicker.SPClientPeoplePickerDict[$('[title^="' + peoplePickerID + '"]')[0].id];
var processedUserList = clientPeoplePicker.ProcessedUserList;
var objectKeys = Object.keys(clientPeoplePicker.ProcessedUserList);
for(var i = 0; i < objectKeys.length; i++) {
if(processedUserList[objectKeys[i]].hasOwnProperty("DisplayName")) {
var displayName = processedUserList[objectKeys[i]].DisplayName;
console.log(displayName);
}
}