2 added 433 characters in body

Bjorn already mentioned one important point: We can define a partial order in any way we want. It should also be mentioned that Cohen defined the order in the other direction: He considered the empty condition the smallest and defined everything else accordingly (i.e., just the other way around). Shelah for instance still uses Cohen's original version of forcing.

Considering $\emptyset$ the largest condition in this particular forcing context agrees with the usual order on Boolean algebras . in the following way:
When forcing with Boolean algebras, the elements of the Boolean algebra correspond to truth values of sentences. If a sentence implies another, the truth value of the first sentence is less or equal to the truth value of the second sentence. The empty condition in Cohen forcing is the truth value of all sentences that are true in every forcing extension by Cohen forcing. This is the largest possible truth value in this context. (This is not to be confused with the fact that $\emptyset$ is the smallest element of the Boolean algebra $\mathcal P(\omega)$. The Boolean algebra corresponding to Cohen forcing is very different from $\mathcal P(\omega)$.) This interpretation of the direction of the order agrees with Bjorn's explanation saying that $\emptyset$ leaves open the most possibilities.

Anyhow. The whole discussion above in somewhat unnecessary since your problem is actually at a different level: The partial order that we are talking about is just a technical tool in order to construct a forcing extension. It has little to do with the containment relation on the subsets of $\mathbb N$. In fact, the containment relation among subsets is the same in the ground model (the one that satisfies CH) and the generic extension. I.e., if $a,b\subseteq\mathbb N$ are both in the ground model, then $a\subseteq b$ holds in the extension iff it holds in the ground model. In the extension we just have many more subsets of $\mathbb N$.

1

Bjorn already mentioned one important point: We can define a partial order in any way we want. It should also be mentioned that Cohen defined the order in the other direction: He considered the empty condition the smallest and defined everything else accordingly (i.e., just the other way around). Shelah for instance still uses Cohen's original version of forcing. Considering $\emptyset$ the largest condition agrees with the usual order on Boolean algebras. When forcing with Boolean algebras, the elements of the Boolean algebra correspond to truth values of sentences. If a sentence implies another, the truth value of the first sentence is less or equal to the truth value of the second sentence. The empty condition in Cohen forcing is the truth value of all sentences that are true in every forcing extension by Cohen forcing. This is the largest possible truth value in this context.

Anyhow. The whole discussion above in somewhat unnecessary since your problem is actually at a different level: The partial order that we are talking about is just a technical tool in order to construct a forcing extension. It has little to do with the containment relation on the subsets of $\mathbb N$. In fact, the containment relation among subsets is the same in the ground model (the one that satisfies CH) and the generic extension. I.e., if $a,b\subseteq\mathbb N$ are both in the ground model, then $a\subseteq b$ holds in the extension iff it holds in the ground model. In the extension we just have many more subsets of $\mathbb N$.