From what I know, one of the strategies employed by an SVM is to increase dimensionality of your data until they are linearly separable. (I guess there's some mathematical proof that your data will always eventually be linearly separable in some high dimensional space)
So for example, if you have two attributes $x$ and $y$, and you have a function $\phi$ that maps these attributes to higher dimensional ones, then part of the SVM's calculation would be the dot product between $\phi(x)$ and $\phi(y)$.
But $\phi(x)$ and $\phi(y)$ are higher dimensional versions of $x$ and $y$ and thus it is more expensive computation and storage wise to compute the dot product between them. That's where the kernel function comes in: instead of calculating $<\phi(x), \phi(y)>$, you calculate $K(x, y)$ , where $K$ is a valid kernel function.
For example, if we use $K(x,y) = (1+x^Ty)^2$, then $K(x, y) = <\phi(x), \phi(y)>$. But $K$ is much cheaper to compute than the dot product between those high dimensional vectors.
However, I never defined how exactly $\phi$ maps vectors to higher dimensional versions of themselves. Does the exact specification of how the mapping happens not matter?
TLDR $K(x, y) = <\phi(x), \phi(y)>$ is said to hold even when $\phi$ hasn't been defined. Why? Doesn't $\phi$ need to be defined?