Learn French: Nous croyons que cette histoire est vraie. - We believe that this story is true.

The verb croire (to believe) is irregular. For the nous and vous forms, the 'i' in the stem changes to a 'y'. For the other present tense forms, it keeps the 'i', like in je crois or ils croient.

Can I leave out que just like we sometimes leave out the word 'that' in English?

No, in French you must always include que to connect these kinds of phrases. While an English speaker might just say 'We believe this story is true', in French you must strictly say 'We believe THAT this story is true'.

Why do we use cette instead of cet before a noun starting with a mute 'h'?

Histoire (story) is a feminine noun, so it must use the feminine word for 'this', which is cette. The special form cet is only used for masculine nouns that start with a vowel or a mute 'h'.

Why is there an 'e' at the end of vraie?

Adjectives in French must match the gender of the noun they describe. Since cette histoire is feminine, the adjective vrai (true) gets an extra 'e' added to the end to become vraie.