Learn Russian: Он изучает карту.

Because it is the direct object of the sentence (the thing being studied). In Russian, when a feminine noun ending in -а (like карта) is the direct object, it takes the Accusative case and the ending changes to -у. Remember that while masculine and neuter inanimate objects don't change their endings in the Accusative case, feminine nouns always do!

Why use изучает for studying here?

The verb изучать (which becomes изучает for 'he') means to study something closely, examine it in depth, or study an academic subject. This is exactly what you do when you are trying to read and figure out a map! It is different from учиться, which simply means to be a student somewhere.

Where are the words for 'is' and 'the' in this sentence?

Russian doesn't use articles like 'a' or 'the', so карту can mean 'a map' or 'the map' depending on context. Also, Russian has only one present tense. The single verb изучает covers both 'studies' (habitually) and 'is studying' (right now).

Why does the verb end in -ет?

This is a quick refresher on present tense verbs. The dictionary form of this verb is изучать. To make it match the subject он (he), we drop the -ть ending and add the standard 'he/she/it' ending for this group of verbs, which is -ет.