A solution without additional extensions

With sass

Assuming you have sass installed globally with for instance:

npm install -g sass

Open the folder and create a task.json file under .vscode containing

{
// See https://go.microsoft.com/fwlink/?LinkId=733558
// for the documentation about the tasks.json format
"version": "2.0.0",
"tasks": [
    {
        "label": "Watch Sass",
        "type": "shell",
        "command": "sass --watch src/style.sass styles/style.css --style=compressed",
        "problemMatcher": [],
        "group": {
            "kind": "build",
            "isDefault": true
        },
        "runOptions": {
            "runOn": "folderOpen"
        }
    }]
}

With node-sass

Replace sass with node-sass in the above.

In both cases make sure the source/destination filename, location and extension are correct (in my case src/style.scss and style/style.css)

With a Workspace file

Or copy the section in your .vscode-workspace file to avoid clutter of .json files.

Make sure to change the sass source and destination files to your personal needs.

Setup VSCode

[EDIT] whith the current version this is asked the first time you open the workspace file and the following steps are no longer needed. To a llow automatic run tasks

1. Ctrl+Shift+P
2. select Manage automatic Tasks and
3. select Allow Automatic Tasks in Folder and
4. close and reopen your folder (or Workspace)

The sass compiler will be called and starts watching all your edits with a reassuring:

Compiled css\src\style.sass to css\style.css.
Sass is watching for changes. Press Ctrl-C to stop.

or with error messages when compilation failed.:

Error: semicolons aren't allowed in the indented syntax.
  ╷
7 │     padding: 0;
  │               ^
  ╵
  css\src\_base.sass 7:12  @import
  css\src\style.sass 1:9   root stylesheet

EDIT command and args can be separated

{
    "label": "Compile sass",
    "type": "shell",
    "command": "sass",
    "args": [
        "--watch",
        "--style=compressed",
        "./style/src/main.sass",
        "./style/main.css"
    ],
    "runOptions": {
        "runOn": "folderOpen"
    }
},

Browsers do not automatically understand the SCSS files, you need to compile such files first into CSS. If you are a Node.js user, you may install the SASS compiler by running the command:

npm install -g sass

Then compile your SCSS file by running the command:

sass input.scss output.css

Now you can link output.css in your HTML file.

Make sure the index.css file is inside the CSS folder, then add the below link tag.

<link rel="stylesheet" href="./CSS/index.css">