Learn Hardcore Swedish: Många människor volonterar idag.

Because många already works as a determiner, like many in English. In Swedish, after words like många (many), få (few), några (some), and alla (all), you normally do not add an article.

So:

många människor = many people
not många en människor or anything similar

This is very similar to English, where you say many people, not many the people.

What are the singular and plural forms of människa?

The word is:

en människa = a person / a human being
människor = people / human beings

So in this sentence, människor is the plural form because många requires a plural noun.

What does många människor literally mean?

Literally, it means many people.

A word-by-word breakdown:

många = many
människor = people

It is a very normal Swedish noun phrase.