The widgets you see in widgets application within the Admin UI are for portal development just like the web parts. They are not for MVC.
After you have created widget in your MVC project and registered the widget, they will all appear if you add this line into the VIEW .cshtml
@Html.Kentico().EditableArea("area1")
If you want to specify certain widgets to be available to an area, you can do this
@Html.Kentico().EditableArea(
"area2",
allowedWidgets: new[] { "DancingGoat.General.TextWidget", "DancingGoat.HomePage.ImageWidget", "DancingGoat.Widgets.KCCafeWidget", "DancingGoat.General.HTMLWidget" }
)