Essentially Java Applets made Java a success.
Java Applets run in a sandbox but a signed Applet could do just about anything. A security risk, yes, but rather than fixing the security we have just thrown the whole thing out.
Given that nearly all browser makers are already getting rid of plugin support, you can argue that Oracle really didn't have any choice but drop the feature. The argument is that plugins have to connect deeply to the browser's architecture and this makes it too difficult to keep everything up-to-date and secure. In the case of the Java plugin things were even more complex because of the need to use the JVM etc to actually run the code. Keeping all of this software up-to-date was a task beyond most users. So plugins of all types, including Java, had to go.
The only common browser with any sort og ongoing support for plugins is IE 11, which is currently in maintenance only mode. It is ironic that the last browser to support Java is Microsoft's, given the love hate relationship there has been between the two.
The loss of the plugin is going to be a big problem. A lot of software has been written as Java Applets. Banks and big corporations found the ability to use Java on their backend servers and in the client too good to ignore. There are also lots of sophisticated web based programs that rely on Java Applets and these are just going to become unusable.
The suggested upgrade path is to make use of Web Start. This allows you to run a Java application downloaded from the web in a sandbox. It sounds a lot like Applet technology, but the big difference is that it doesn't integrate with the browser. When the user launches a Web Start app, either from the browser or from the desktop, it runs in a separate window. To make use of Web Start the user also has to install a JNLP client - usually the Java Web Start client. This can be done along with launching the first Web Start app, i.e. the user can be given the choice of downloading and installing the JNLP client along with the Web Start app of that they want to run. You can think of the JNLP client as a sort of Java web browser in the sense that it downloads and renders a JNLP file sent by the server in much the same way that a browser renders HTML.
The good news is that, in principle, a JNLP client will just run an existing Applet with no changes. All you need to to is create a JNLP file to launch the Applet and load any resources it might need.
Let us hope that it really is this trouble free and that a lot of existing and still useful code is rescued.