We can make use of lightning:omniChannelStatusChanged event to listen Salesforce Omni-Channel Presence Status changes by the agents. lightning:omniChannelStatusChanged event will be fired when the agents sets or changes their Status in the Omni-Channel widget.

Sample Aura Component:

Component:

<aura:component 
                implements="flexipage:availableForAllPageTypes" 
                access="global" >
    <lightning:omniToolkitAPI 
                              aura:id="omniToolkit" />  
    <aura:handler 
                  event="lightning:omniChannelStatusChanged" 
                  action="{! c.onStatusChanged }"/>
</aura:component>

Controller:

( {
    onStatusChanged : function( component, event, helper ) {
        
        let statusId = event.getParam('statusId');
        let channels = event.getParam('channels');
        let statusName = event.getParam('statusName');
        let statusApiName = event.getParam('statusApiName');
        console.log(
            'statusId is',
            statusId
        );
        console.log(
            'channels are',
            channels
        );
        console.log(
            'statusName is',
            statusName);
        console.log(
            'statusApiName is',
            statusApiName
        );
        
    }
    
} )

Output:

Also, we can use UserServicePresence object/entity to track the Salesforce Omni-Channel Presence Status Changes.

How to find Users status changes in Salesforce Omni-Channel?