Learn Russian: Русская культура очень интересная.

In Russian, the present tense of the verb "to be" is almost always left out. Instead of using a word for "is", you simply place the subject and its description side by side: очень интересная (very interesting).

Why do both русская and интересная end in -ая?

Both words are adjectives describing the noun культура (culture). Since культура is a feminine singular noun in the nominative (subject) case, any adjectives describing it must also take the feminine nominative ending, which is typically -ая.

Could I use the short-form adjective интересна here instead of интересная?

Yes, you absolutely can! When an adjective describes a subject after an implied "is" (as a predicate), Russian often uses the short form. Saying Русская культура очень интересна is completely correct, though it sounds a bit more formal or literary. The long form интересная used in this sentence is very natural for everyday conversation.