If $B$ is a Boolean algebra with $\bot$ as its bottom element (your $F$), an element $a\in B$ is an atom of $B$ if $a\ne\bot$ and $\{b\in B:\bot. In other words the atoms of $B$ are the minimal non-$\bot$ elements.
You can’t show that $D_{30}$ by itself is a Boolean algebra, but you can show that it becomes one when you give it the right operations and order. The join of two elements of $D_{30}$ is their least common multiple, and the meet is their greatest common divisor: $m\land n=\gcd\{m,n\}$ and $m\lor n=\operatorname{lcm}\{m,n\}$. This means that the lattice order on $D_{30}$ is the divisibility relation: if $\preceq$ denotes the lattice order ($m\preceq n$ iff $m\land n=m$), then $m\preceq n$ iff $m\mid n$. It’s not hard to see that this means that $\bot=1$ and $\top=30$. The minimal elements of $D_{30}\setminus\{1\}$ are then $2,3$, and $5$: every other member of $D_{30}\setminus\{1\}$ has a non-trivial divisor and therefore fails to be minimal.
To show that $D_{30}$ with these operations really is a Boolean algebra, you simply have to verify that the operations have the required properties: each distributes over the other, and the lattice is complemented. The only part of this that perhaps isn’t completely routine is figuring out what the complement of $n\in D_{30}$ is. You need an element $m$ such that $n\land m=\bot$ and $n\lor m=\top$, i.e., such that $\gcd\{n,m\}=1$ and $\operatorname{lcm}\{n,m\}=30$; what’s a simple description of that $m$ in terms of $n$?