Learn Russian: Я знаю твоего старшего брата, но у меня нет его номера. - I know your older brother, but I do not have his number.

Because they describe a masculine animate object (a living being). In the Accusative case, which we use for the direct object of "I know", masculine nouns and their adjectives take the exact same endings as they do in the Genitive case. The dictionary forms твой and старший change to твоего and старшего to match the animate noun брата.

How do I pronounce the "-его" endings in твоего and старшего?

The letter г in these specific modifier endings is pronounced like an English "v". So, твоего sounds like "tvoyevo" and старшего sounds like "starshevo". This pronunciation rule applies whenever you see "-ого" or "-его" as an adjective or pronoun ending.

Is the word его in the second half of the sentence related to those "-его" endings?

Yes! The word его (meaning "his") is actually the Genitive form of он (he). To say "his number", Russians literally say "the number of him". Unlike adjectives, this possessive word его is frozen and never changes its ending, no matter what case the noun after it is in.

Why does the sentence say номера instead of just номер?