How to know tomcat server: JSP page is modified ..?

Because when Tomcat is asked to execute a JSP, is compares the modification date of the JSP file with the modification time of the compiled class corresponding to this JSP, and if more recent, it recompiles on the fly before executing it.

This is BTW an option that should be turned off in production, because it takes time to perform this check.

See http://tomcat.apache.org/tomcat-7.0-doc/jasper-howto.html for details.

It is upto the container to decide when to load servlets. A servlet can be loaded at runtime on demand. And coming to JSP, JSP translated to servlet can also be loaded at runtime.

Coming to your question,

Why Tomcat does not require restart?

It is because Tomcat is capable of adding/modifying classpath to Web Application classloader at runtime. Tomcat will be having their custom Classloader implementation which allows them to add the classpaths at runtime.

How does the custom classloader might work?

One way to get this working is when a Servlet/JSP is modified, a new classloader is created for the Servlet/JSP with Application classloader as parent classloader . And the new classloader will load the modified class again.