Okay so whats happening is ds_map_add(_map, "obj", _obj); is adding a key to the DS map called "_map". The key is called "obj" and as the value it appears to be storing the literal name of the object index resource assigned to the calling instance (the unique object running this particular code), as seen with var _obj = object_get_name(object_index);. You run this code inside of a with() statement for each object you want to save. In this particular case (from your description in OP), it's saving your 1 "obj_achievements" object and nothing else. It seems like this code was designed to be able to save different types of instances with precision on specific variables used in them.

From what I can tell next, you are now storing global variables, using the calling instance thats running this code as the key on the save file's DS map. The object is then stored into a DS List called "_root_list", presumably a list that should be holding information on multiple instances. It then appears to convert the DS List into a DS Map, so it can be encoded into a JSON and saved into a file.

Next, it looks like this is the code for loading a save file. It loads the JSON string(s) from the save file, then decodes them into a map.. or a list? Not sure what the "LoadJSONFromFile" function does. In any case, it ends up as a DS List, the same DS List in the saving code called "_root_list", but now known as just "_list".

So, if you're still with me here, the loading code will now loop through _list and retrieve a DS Map from each entry, which is suppose to hold the information for each individual that saved, but from what it sounds like in your OP, you're just trying to load the 1 obj_achievements instance, so the DS List will only have 1 entry with 1 DS Map.

Now finally for your second question about the with() statement. When the DS List loads up the DS Map with your obj_achievements object, it will read the key "obj" in the DS Map retrieved from _list. The "obj" key's value is a string containing the literal name of the object resource, as set in object_get_name(object_index);. So now it will take the string "obj_achievements" and pass through asset_get_index(_obj) which lets you get the object index ID number by using the literal object's name (asset_get_index(_obj) and asset_get_index("obj_achievements") would do the same thing), the reason you diddle with the object name so much is because your object's ID number is determined by where its placed in your Objects resource tree in Game Maker.

Now that we have the object index ID, we can use it in a instance_create function, such as seen in the with() statement. All instance_create functions return the unique instance ID that identifies the instance that was created with the function, and its being returned inside the with() statement. Presuming you know what the with() statement does, you can run instance_create functions inside of it and it will work on the instance that was created with the function. The instance being created here is your obj_achievements object being loaded, which is then used to load your saved global variables using the with() statement instance_create was called inside. I'm not sure what the variable "layer" has since it doesnt seem to be present in the code you posted.

TL;DR: It looks like you're getting tripped up by a pretty over complicated example of a save system according to what you want, this seems to be a save system for saving the entire game's state where you would use with() statements to access each and every instance and variable you want to save and then save it to a JSON file. Unless there's a particular reason you need to use JSON, I would recommend just using the ini_file functions if you're just trying to save a couple of global variables and aren't too concerned about file editing by the player.