Learn Italian: Loro sono cugini.

The verb essere (to be) uses the exact same word, sono, for both 'I am' (io sono) and 'they are' (loro sono). You can tell which one is meant by the subject pronoun or the context. Since cugini is plural, the verb has to mean 'they are'.

If cugini is masculine plural, does that mean they are all male cousins?

Not necessarily. In Italian, the masculine plural is used for a group of males, but also for a mixed group of males and females. So cugini can be just men, or a mix. If it were a group of only female cousins, you would say cugine.

Can I drop the loro and just say Sono cugini?

Yes, you can! Italian often drops the subject pronoun. Even though sono can mean 'I am', the plural noun cugini makes it obvious that you mean 'They are cousins' and not 'I am cousins'.