Learn Croatian: Stanem i ugledam ogromnog vepra. - I stop and catch sight of a huge boar.

This is the Historical Present. In Croatian, you can use the present tense of perfective verbs to tell a story about a past event. It makes the narrative feel sudden, vivid, and immediate, like you are reliving the moment right now. In English, you might say 'I stop and catch sight...' to get the exact same dramatic effect.

I thought stati meant 'to stand'. Why is stanem translated as 'I stop' here?

Stati is a versatile perfective verb. While it does mean 'to stand' or 'to step', it is also the standard verb for 'to stop' in the sense of halting your physical movement. If you wanted to say 'to stop doing an action', you would use a different verb: prestati.

Why is the object ogromnog vepra instead of ogroman vepar?

Because a boar is a living creature. Vepar is a masculine animate noun, and because it is the direct object of the sentence, it must be in the accusative case. For masculine animate nouns, the accusative endings look exactly like the genitive case endings, which gives us the -og ending on the adjective and the -a ending on the noun.

Could I say gledam instead of ugledam in this sentence?