Inside a composite fairing is an adverbial phrase here. It specifies the manner in which the payloads will be delivered. If you replaced composite fairing with rocket, the grammar would be the same (though not the semantics, obviously), but hopefully the simplified example makes it easier to understand.
A fairing is a piece of a rocket designed to shield cargo from the atmosphere during launch. Here's a brief synopsis from the Wikipedia article:
A payload fairing is a nose cone used to protect a spacecraft (launch vehicle payload) against the impact of dynamic pressure and aerodynamic heating during launch through an atmosphere.
Composite is an adjective modifying fairing, and it means that it's constructed from multiple materials.
The entire phrase refines the meaning of the verb deliver by telling us the precise manner of action. Damkerng T. offers this example sentence with parallel grammar:
This time James Bond will carry his high tech gun inside his suitcase.
Here, inside his suitcase serves the same grammatical function as inside a composite fairing; it tells us how James bond will carry his high tech gun.
Can I use in a composite fairing instead?
Yes, you can replace inside with in and the meaning will still be essentially the same. Inside is the better choice, though, because it's more specific than in, and in this case, the payload is literally enclosed within the fairing. In doesn't entail enclosure but inside does. Take a look at this question for more on the nuanced differences between these two prepositions.