First, Maps are defined with two data types, such that we would express this in generic terms written with a unique letter for each data type involved: Map<T, U>, or as the documentation states: Map<T1, T2>. I prefer the classic nomenclature, but choose whichever you feel you understand better.

In your case, T is Id, and U is Map<T′, U′>. T′ and U′ are the type String. If you remember that, it should be easier to construct complicated types in the future:

Map< T ,            U          >
Map< Id, Map<   T′  ,   U′   > >
Map< Id, Map< String, String > > 

Second, to populate nested structures, you have to put them inside each other. In other words, before you can populate a value in the inner map, the outer map must have the object created first. To create most objects, we use the new keyword. One new is one new object.

When we create a new object, we call a constructor, which can take zero or more parameters to configure the initial state of the object. They are normal methods, which means we need to pass in any parameters, like we would any other method. new Map<Id, Map<String, String>>(), for example, constructs a new object with the compound data type Map<Id, Map<String, String>>.

So, to populate such a Map, we would do something like:

Map<Id, Map<String, String>> myMap = new Map<Id, Map<String, String>>();
myMap.put(obj.Id, new Map<String, String>());
myMap.get(obj.Id).put('value1','value2');

Note that put displaces the previous value if it exists, so we normally check first:

Map<String, String> myMapValue = myMap.get(obj.Id);
if(myMapValue == null) {
  myMapValue = new Map<String, String>();
  myMap.put(obj.Id, myMapValue);
}
myMapValue.put('value1','value2');

You can initialize a Map with the {} constructor format:

Map<Id, Map<String, String>> myMap = new Map<Id, Map<String, String>> {
  obj.Id => new Map<String, String> {
    'value1' => 'value2'
  }
};

Notice how there are two new keywords. We are still creating new objects, just placing one inside the other initially. The => operator simply means "key => value." This is the only use of => in Apex at this time.

Note that putAll is used to put a bunch of values inside the map at once. It requires that the parameter provided is the same type as U from our example above.

Map<Id, Map<String, String>> myMap = new Map<Id, Map<String, String>>();
Map<String, String> params = new Map<String, String>();
params.put('value1','value2');
myMap.put(obj.Id, new Map<String, String>());
myMap.get(obj.Id).putAll(params);

Using your sample code will work, though of course the second level map can only ever have one entry, if you're in a situation where you may need more then the code below will do the trick for you:

    // This line creates the map in the proper format
    Map<Id,Map<Id,Addendum__c>> addendums = new Map<Id,Map<Id,Addendum__c>>{};
    
    // This for loop goes through each addendum and first places it into a single dimension map.  
    // Once that map is created, it is placed into the final multi-dimensional
    for(Addendum__c addendum : [SELECT Id,Opportunity__c FROM Addendum__c WHERE Opportunity__c IN :oppBatch])
    {
        if(addendums.get(addendum.Opportunity__c) == null)
        {
            addendums.put(addendum.Opportunity__c, new Map<Id, Addendum__c>{addendum.Id => addendum);
        }
        else
        {
            addendums.get(addendum.Opportunity__c).put(addendum.Id, addendum);
        }
    }

As you can see this uses get() on the first map to get the correct second-level map in which we want to add an addendum.

Use myMap.values() to "flatten" the map into a List (in your case list of accounts), insert the whole list. Such flattening just keeps references so the original accounts in the map will be silently updated with the generated record Id.

Map<String, Account> accounts = new Map<String, Account>{
    'A13' => new Account(Name ='Test'),
    'B14' => new Account(Name = 'Another')
};
insert accounts.values(); // You can use Database.insert() if you want, it accepts a list too.

Map<String, Id> idsByKey = new Map<String, Id>();
for(String s : accounts.keyset()){
    idsByKey.put(s, accounts.get(s).Id);
}
System.debug(idsByKey);

If you want to add a map onto existing map, use putAll() method.

finalMap.get('DETAILS').putAll(subDetails); 

It will add the entries of subDetails onto whatever Map finalMap contains aganist key 'DETAILS'.

Replace

counts.put('month_total',counts.get['month_total']++);

with

counts.put('month_total',counts.get('month_total')++);