Learn Russian: Недавно они были на празднике.

Были is the plural past tense form of the verb быть (to be). Because the subject is the plural pronoun они (they), the verb must match it. If the subject were a single masculine person, it would be был, and for a feminine person, была.

Why does this sentence use на instead of в for "at"?

In Russian, when you are talking about attending an event, gathering, or activity (like a celebration, concert, or meeting), you use the preposition на instead of в. You can think of it as being "at" an experience rather than physically "inside" a building.

Why does the word праздник end with an -е here?

Because it follows the preposition на to talk about a location (answering the question "where?"), the noun must be in the Prepositional case. The dictionary form is праздник, which is a masculine noun ending in a consonant. To form the Prepositional case, we just add an -е to the end, making it празднике.

I noticed there is a д in празднике. Do I pronounce it?

No, the д is completely silent! You should pronounce it as if it were written "празник". In Russian, when certain consonants bunch together (like the здн combination here), the middle one is often dropped in speech to make the word easier to say.