The question asked 'what exactly operation "+=" do in map ??' is not relevant to the situation you described. Map's method += adds entry (key -> value) to Map, so that map(key) == value.
So the question is: what does map(1) += 10 does.
This simple lines
map(1) += 10
compiles into this javacode
24: invokestatic #36 // Method scala/runtime/BoxesRunTime.boxToInteger:(I)Ljava/lang/Integer;
27: aload_2
28: iconst_1
29: invokestatic #36 // Method scala/runtime/BoxesRunTime.boxToInteger:(I)Ljava/lang/Integer;
32: invokeinterface #43, 2 // InterfaceMethod scala/collection/mutable/Map.apply:(Ljava/lang/Object;)Ljava/lang/Object;
37: invokestatic #47 // Method scala/runtime/BoxesRunTime.unboxToInt:(Ljava/lang/Object;)I
40: bipush 10
42: iadd
43: invokestatic #36 // Method scala/runtime/BoxesRunTime.boxToInteger:(I)Ljava/lang/Integer;
46: invokeinterface #51, 3 // InterfaceMethod scala/collection/mutable/Map.update:(Ljava/lang/Object;Ljava/lang/Object;)V
Remove boxing and unboxing ops - we are not interested in them(line 24, 29, 37, 43). Review the rest:
We get value that was in map
27: aload_2
28: iconst_1
32: invokeinterface #43, 2 // InterfaceMethod scala/collection/mutable/Map.apply:(Ljava/lang/Object;)Ljava/lang/Object;
Add 10 to it:
40: bipush 10
42: iadd
And accomplish update:
46: invokeinterface #51, 3 // InterfaceMethod scala/collection/mutable/Map.update:(Ljava/lang/Object;Ljava/lang/Object;)V
Seems map(1) += 10 is desugared to map.update(1, map.apply(1) + 10)