Learn Portuguese (Portugal): Conta-me tudo!

Because this is a direct command. When giving an affirmative command to someone you address as tu (informal 'you'), you drop the final 's' from the present tense. So, tu contas (you tell) becomes conta (tell!).

Why is the pronoun me attached to the end of the verb with a hyphen?

In European Portuguese, object pronouns like me normally attach to the end of the verb in an affirmative command. You will almost never see a sentence start with an object pronoun like me in Portugal, so Conta-me is the correct structure.

I thought contar meant 'to count'. Can it also mean 'to tell'?

Yes! Just like in English where you might 'recount' a story, the verb contar means both to count numbers and to tell a story, a secret, or a piece of news. In this sentence, it clearly means 'to tell'.

Why use tudo here and not todo?

Tudo means 'everything' as a broad, general concept, and it never changes form. You use words like todo or todos when you are talking about specific nouns (like 'all the books' or 'everyone'). Because you want to hear 'everything' in general, tudo is the exact right word.