So in an Arrow-Debreu market, there is a price for every asset/claim for every state of the world. The states $s$ refer to this.
For your utility maximization problem, you are maximizing a combination of what you consume, $u(c)$, and the expected profit from your portfolio of claims, which can then be put into future consumption.
$\pi(s)$ describes the profit in state $s$, and that is multiplied by the utility of consumption in that state $s$. In this case, I guess the maximizer does not seem to worry about variance, so maximizing sum of those possible outcomes in all states of the world suffices. Finally, $\beta$ represents the fact that you prefer to consume immediately instead of investing and waiting to see the state of the world.
So the budget constraint can be roughly described as such: whatever you decide to consume now plus the cost of buying claims that will give you money to consume later must equal whatever money you start with plus whatever money you get from buying those cool claims.