what is a 1-bit PRF?
That would usually be a PRF with a 1-bit output, but the rest of the text clarifies it is meant a 1-bit input.
what is the $\{0,1\}$ in $F:\ K \times \{0,1\} \to K$ ?
It is the set with two elements $0$ and $1$. This also is the set of the possible values of the second/right/normal/non-key component of the input of the PRF $F$.
Note: a Pseudo Random Function Family (confusingly abbreviated PRF just like a Pseudo Random Function) has two inputs (equivalently: an input consisting of an ordered pair). The first (the left component of the pair) is the key, and the second (the right one) is the normal/non-key input (which is the single input of any particular member of the Pseudo Random Function Family). The member of the Pseudo Random Function Family corresponding to key $k$ is often noted $F_k$. By that notation $F_k(x)=F(k,x)=F((k,x))$. Here, $F_k:\ \{0,1\}\to K$ (that's a particular Pseudo Random Function for a certain key $k$), since $F:\ K \times \{0,1\} \to K$ (that's the whole Pseudo Random Function Family).
what does $G(k)[x]$ in $F(k, x\in\{0,1\}) = G(k)[x]$ mean?
The PRG $G$ takes an input $k$ and produces an output $G(k)$ consisting of two parts of the same width as $k$, as apparent in $G:\ K \to K^2$. This is the same as $G:\ K \to K\times K$. That is, the output of $G$ is an ordered pair of bitstrings, assimilated to a bitstring twice as wide as $k$, that we split in two parts as wide as $k$, each from the same set $K$ as $k$ is from.
$G(k)[x]$ is the first or the second component/part of the output of $G(k)$, according to $x$ being $0$ or $1$. It's also the output of $F$ for input $(k,x)$ where $k$ is the PRF's key, and $x$ is the PRF's second/right/normal/non-key input, which is a single bit.
What exactly is $x$?
It's a bit. It's not an input of the PRG $G$. It selects among the two halves of the output of the PRG $G$ for input $k$, that is $G(k)$. We can think of $[x]$ as we'd do for [x] in a computer language with indexes.
