Learn Finnish: Opit kieltä puhumalla aktiivisesti muiden ihmisten kanssa. - You learn the language by speaking actively with other people.

This is the third infinitive (often called the MA-infinitive) in the adessive case. You form it by taking the strong verb stem (puhu-), adding -ma- or -mä-, and then the adessive ending -lla or -llä. This specific -malla structure is used to express how or by what means an action is done.

Why is the word for language (kieli) in the partitive case (kieltä) here?

The verb oppia (to learn) takes a partitive object when the learning is an ongoing, continuous process. Since learning a language isn't something you finish in a single instant, kieltä is in the partitive to reflect that the action is not yet complete.

Why are the words muiden ihmisten in the genitive case?

They are paired with the postposition kanssa (with). In Finnish, the word that comes just before the postposition kanssa must always be in the genitive case. Because this sentence is about plural people, both muu (other) and ihminen (person) are in the plural genitive form.

How do we get the form muiden from the basic word muu (other)?

To form the plural, the double vowel at the end of muu drops one letter to make room for the plural marker -i-, giving the plural stem mui-. Then, the standard plural genitive ending -den is added, resulting in muiden.