What exactly does Keyspace $K^3$ mean? This is from Dan Boneh's book
4.5 Constructing block ciphers from PRFs
Let F be a PRF, defined over $(K, X, X)$, where $X = \{0, 1\}^n$. We describe a block cipher $\varepsilon = (E, D)$ whose key space is $K^3$, and whose data block space is $X^2$.
Given a key $(k1, k2, k3) \in K^3$ and a data block $(u, v) \in X^2$, the encryption algorithm E runs as follows:
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What exactly is  $K^3$ &amp; $X^2$ here? Why has the KeySpace been cubed &amp; the datablock space been squared?
What exactly are $K^3$ and $X^2$ here?

The notation $K^3$ where $K$ is the set of keys $k$ stands for $K\times K\times K$, that is the set of triplets $(k_1,k_2,k_3)$ with $k_1\in K$, $k_2\in K$, $k_3\in K$. When $K=\{0,1\}^n$ (the set of $n$-bit bitsrings), we can assimilate $K^3$ to $\{0,1\}^{3n}$ (mathematicians do this when there's a trivial bijection between two sets; here that's concatenation of bitstrings in one direction, and splitting into 3 equaly-sized bitstrings in the other).
Similarly the notation $X^2$ where $X$ is the set of blocks $x$ stands for $X\times X$, that is the set of pairs $(x_1,x_2)$ with $x_1\in X$, $x_2\in X$. When $X=\{0,1\}^b$, we can assimilate $X^2$ to $\{0,1\}^{2n}$.

Why has the key space been cubed and the block space been squared?

That must be the objective of the skipped construction. Increasing the keyspace is easy and there's one common example: 3DES cubes or squares the keyspace, leaving the blockspace unchanged. Increasing the blockspace is slightly trickier.
The cardinality of the key space is cubed: it goes from $2^n$ to $2^{3n}=\left(2^n\right)^3$. The cardinality of the block space is squared: it goes from $2^b$ to $2^{2b}=\left(2^b\right)^2$.