The simple answer is because it is plural. Das große blaue Pferd becomes die großen blauen Pferde. Sein takes the place of das (possessive determiner as others have said) and so Sein große blaue Pferd becomes Seine großen blauen Pferde. The other comments are correct, it is mixed inflection but the easy way to remember it is that after Die or the possessive determiners, all descriptive adjectives in the plural end with n or en.
After possessive determiners (e.g. "seine"), the mixed inflection is used.
Take a look at Hartmut Rastalsky's site at UMichigan. Unique and simple: (or at least as simple as German gets :)
If you go to the "Colors" lesson and click on Tips and Notes, then scroll down to the Nominative section, you'll see under plural how there is an option with define article, indefinite article and no article.
So, in the case of your question, "seine" would fit under "keine" above, which means it gets an n at the end of grosse(n).
Had there been no "Seine" at the beginning of the sentence it would have been written "Grosse Tiere haben Hunger."
Hope that helps.
Is "grossen" incorrect or am I missing something. When there is an adjective and no article I thought adjectives of plurals that are nominative case would take just an e at the end (http://german.morley-computing.co.uk/adjectiveEndings3.php)
@kevinbyberge : sein (possessive pronoun) always means "his"
See full inflection here: http://canoo.net/inflection/sein:Pron:Poss:3rd:M:SG