What are '$$' used for in PL/pgSQL

These dollar signs ($$) are used for dollar quoting, which is in no way specific to function definitions. It can be used to replace single quotes enclosing string literals (constants) anywhere in SQL scripts.

The body of a function happens to be such a string literal. Dollar-quoting is a PostgreSQL-specific substitute for single quotes to avoid escaping of nested single quotes (recursively). You could enclose the function body in single-quotes just as well. But then you'd have to escape all single-quotes in the body:

CREATE OR REPLACE FUNCTION check_phone_number(text)
  RETURNS boolean
  LANGUAGE plpgsql STRICT IMMUTABLE AS
'
BEGIN
  IF NOT $1 ~  e''^\\+\\d{3}\\ \\d{3} \\d{3} \\d{3}$'' THEN
    RAISE EXCEPTION ''Malformed string "%". Expected format is +999 999'';
  END IF;
  RETURN true; 
END
';

This isn't such a good idea. Use dollar-quoting instead. More specifically, also put a token between the $$ to make each pair unique - you might want to use nested dollar-quotes inside the function body. I do that a lot, actually.

CREATE OR REPLACE FUNCTION check_phone_number(text)
  RETURNS boolean  
  LANGUAGE plpgsql STRICT IMMUTABLE AS
$func$
BEGIN
 ...
END
$func$;

See:

• Insert text with single quotes in PostgreSQL

As to your second question:
Read the most excellent manual on CREATE FUNCTION to understand the last line of your example.

The $$ is a delimiter you use to indicate where the function definition starts and ends. Consider the following,

CREATE TABLE <name> <definition goes here> <options go here, eg: WITH OIDS>

The create function syntax is similar, but because you are going to use all sorts of SQL in your function (especially the end of statement ; character), the parser would trip if you didn't delimit it. So you should read your statement as:

CREATE OR REPLACE FUNCTION check_phone_number(text)
RETURNS boolean AS <code delimited by $$> LANGUAGE plpgsql STRICT IMMUTABLE;

The stuff after the actual definition are options to give the database more information about your function, so it can optimize its usage.

In fact, if you look under "4.1.2.4. Dollar-Quoted String Constants" in the manual, you will see that you can even use characters in between the dollar symbols and it will all count as one delimiter.