From Cambridge, "whip up someone/something" is a phrasal verb meaning "to cause or encourage a strong feeling or reaction in someone or something". Merriam-Webster defines it as "to excite (someone or something): to cause (someone or something) to feel strong emotions about something".
"Following" here is a noun meaning people. From Cambridge, a "following" is "a group of people who admire something or someone; a group of people who support, admire, or believe in a particular person, group, or idea."
She has attracted a large following among the rich and famous.
The shop has a small but loyal/devoted following.
In this case, it means to gather followers/supporters.
Aberforth here is essentially referring to Dumbledore's (and there is another person here - I forgot) attempts to build up a "following" (or gather followers/supporters) that will help further their beliefs and ideology.
How do they do that? They do it by making clever speeches.
Give yourself a treat. Get yourself a new pair of shoes. Whip yourselves up a following.
If you simplify this, it means "get/build yourselves a following". Or you could say "whip up a following for yourselves".
When you are giving a speech, with the intention to gather followers, you are in that moment. Your words are powerful. There is a lot of emotion in what you say and in your body language. You, yourself, are excited. And that energy spreads across, getting others excited.