Convert the AggregateResult to a List<String>:

public class PickListHandler {
    @AuraEnabled
    public static List<String> getLevel1(){
        List<AggregateResult> groupedLevel1 = [SELECT Level_1__c, COUNT(id) FROM Case_Type_Data__c GROUP BY Level_1__c];
        List<String> level1Values = new List<String>(); // New list to store strings
        for(AggregateResult ar : groupedLevel1){
            // Explicitly cast to String
            level1Values.add(String.valueOf(ar.get('Level_1__c'))); 
            System.debug('Level Value Is' + ar.get('Level_1__c')); 
        }
        return level1Values; // Return the list of strings
    }
}