ASP.Net MVC was designed to give you complete control over the HTML and js, rather than having the framework render these for you, as with ASP.Net. So, nothing really "replaces" the ClientScriptManager. As a result, it is entirely up to you how you handle your js.
You can do something simple like include <script ...>
tags to reference inline script or js files in your _Layout.cshtml
file, or you could use some sort of JavaScript Loader like RequireJS or come up with your own scheme entirely using "Html Helpers" or something.
MVC 4 introduced the concept of bundling which lets you define collections of scripts as "bundles" and have MVC automatically minify and merge them together for you when you reference them in your Views like this :
@Scripts.Render("~/bundles/jquery")