Learn Croatian: Mogu igrati nogomet.

In Croatian, you usually don't need to say the pronoun ja (I) because the verb form already tells you who is doing the action. Mogu specifically means I can. Adding ja mogu would sound unnatural unless you are trying to emphasize it, like saying I can even if someone else cannot.

Why is the verb igrati left in its basic dictionary form?

Igrati is in the infinitive (the basic to play form). When you use a modal verb like mogu (I can), the second verb that follows it stays in the infinitive. This is very similar to English, where we say I can play and not I can playing.

Nogomet is the object of the sentence, so why hasn't its ending changed?

You are right that direct objects take the accusative case. However, nogomet is a masculine noun that refers to a non-living thing. In Croatian, masculine non-living nouns look exactly the same in the accusative object position as they do in their basic dictionary form.

Does nogomet refer to American football or soccer?

Nogomet means soccer. It is made from the words for foot (noga) and movement or flight (met). If you want to talk about American football, you would specify by saying američki nogomet.