What does $\text{dim}_K L$ mean when $K,L$ are fields? +Example. What does $\text{dim}_K L$ when $K,L$ are fields?
From the context in which I have seen it used I believe it means the dimension of the vector space we get from viewing $L$ as a vector space over $K$ and that $\text{dim}_K L=[L:K]$ but I would like some confirmation.
In the example it gives a set $M=\{a+b\sqrt 2~|~a,b \in \Bbb{Q}\}$ and it says $[M:\Bbb{Q}]=2$ would this just be because $\Bbb{Q} \subseteq M$ when we view $M$ as a vector space over $\Bbb{Q}$ we can simply find a basis for $M$ one such basis would be $\{1,\sqrt 2\}$ so then the dimension of $M$ is $2$ so $[M:\Bbb{Q}]=2$?
Finally it says since $\pi$ is transcendental  so $\{1,\pi,\pi^2,\pi^3,...\}$ are linearly independent over $\Bbb{Q}$ so we must have $[\Bbb{R}:\Bbb{Q}]=\infty$ why is this the case could anyone elaborate please?

All of what you said is correct except the last part. You wrote correctly  that $\{1,\pi,\pi^2,\pi^3,...\}$ are linearly independent over $\mathbb Q$ since $\pi$ is trascendental. So $\mathbb R$ contains an infinite set of $\mathbb Q$-linearly independent vectors, therefore $\mathrm{dim}_{\mathbb Q}\mathbb R$ is infinite. But it is not true that  $\{1,\pi,\pi^2,\pi^3,...\}$ is a base of $\mathbb R$ over $\mathbb Q$. In fact, it is not true that any real number is a finite combination with rational coefficients of those elements. If it were so, $\mathbb R$ would be countable (see comment of tomasz).
