Alright a couple things I would change about this. I would get rid of the nested soql statement in your for loop. You could add it in a set first. The variable needs to be inside the nested for loop to be added.

trigger Rollup on Submission__c (after insert) {
   List<Employee__c> Emp=new List<Employee__c>();
   //add a set to get rid of the nested soql in for loop
   Set<String> submittedIds = new Set<String>();
   integer count=0;
  if(trigger.isinsert){
   for(Submission__c S:trigger.new){
      submittedIds.add(S.Submittedby_del__c);
   }
      
      for(Employee__c e:[select id,Address__c,Count__c,Organization__c from Employee__c where id IN :submittedIds])
          //did you try this
          //e.Count__c++:
     
          e.Count__c = e.count__c+1;
     //this was the change you needed.  
     Emp.add(e);                                  
   }
   

update Emp;
}

 I think you are trying to do a rollup summary of how many submissions an employee record has submitted...if I were doing this it would like more like this...this is an example from my developer org with objects related by a lookup relationship and not a master detail

trigger AttendeeTrigger on Attendees__c (after insert,after update) {
    //create set to hold conferences that should be updated
    Set<Id> conferenceIds = new Set<Id>();
    //list of conferences that are updated
    List<Conference__c> conferencesToUpdate = new List<Conference__c>();
    
    
    //populate the set
    for(Attendees__c att : newMap.values()){
        conferenceIds.add(att.Conference__c);
    }
    
    
    //use soql for loop
    for(Conference__c conf : [SELECT Id,Current_Number_Of_Attendees__c,(SELECT Id FROM Attendees__r) FROM Conference__c WHERE Id IN :conferenceIds]){
        conf.Current_Number_Of_Attendees__c = conf.Attendees__r.size();
        conferencesToUpdate.add(conf);
    }
    
    
    //dml to update all conferences at once
    update conferencesToUpdate;

}

 Yours would look something like this

trigger Rollup on Submission__c (after insert) {
    //create set to hold employees that should be updated
    Set<Id> employeeIds = new Set<Id>();
    //list of employees that will be updated
    List<Employee__c> employeesToUpdate = new List<Employee__c>();
    
    
    //populate the set
    for(Submission__c s : Trigger.new){
        employeeIds.add(s.submittedby_del__c);
    }
    
    
    //use soql for loop
    for(Employee__c e : [SELECT Id,Count__c,(SELECT Id FROM Submissions__r) FROM Employee__c WHERE Id IN :employeeIds]){
        e.Count__c = e.submissions__r.size();
        employeesToUpdate.add(e);
    }
    
    
    //dml to update all employees at once
    update employeesToUpdate;

}

 I didn't test this in an ide...so there may be some minor errors but that is the basics of creating a rollup summary on a lookup field for the count of an object. Hope that helps.