For arbitrary subspaces $U,V$ of an arbitrary vector space $W$, we have the dimension formula
$$\dim U + \dim V = \dim (U+V) + \dim (U\cap V).$$
(Take a basis $B_1$ of $U\cap V$, and extend it by systems $B_2$ to a basis of $U$, and $B_3$ to a basis of $V$. Then $B_1 \cup B_2 \cup B_3$ is a basis of $U+V$.)
That yields the
$$\dim F + \dim F^\perp = \dim (F + F^\perp) + \dim (F\cap F^\perp)$$
part. It remains to see that $\dim F + \dim F^\perp = \dim E$. Let's denote the non-degenerate bilinear form by $\beta$. $\beta$ induces a linear map $\Phi \colon E \to E^\ast$ ($E^\ast$ is the dual space of $E$, the space of all linear maps $E\to K$) via
$$\Phi(x)(y) = \beta(x,y).$$
The non-degeneracy of $\beta$ is equivalent to the injectivity of $\Phi$. Since the spaces are finite-dimensional, we have $\dim E^\ast = \dim E$, and thus $\Phi$ is an isomorphism. For a subspace $F \subset E$, the image of $F^\perp$ under $\Phi$ is the annihilator of $F$,
$$\Phi(F^\perp) = F^0 = \{ \lambda \in E^\ast : F\subset \ker\lambda\}.$$
If you already know that $\dim F + \dim F^0 = \dim E$ for all subspaces of finite-dimensional spaces, that's it. Otherwise, to see that, choose a basis $B_0 = \{v_1,\dotsc, v_f\}$ of $F$, extend it to a basis $B = B_0 \cup B_1 = \{v_1,\dotsc,v_f,v_{f+1},\dotsc,v_e\}$ of $E$, and consider the dual basis $B^\ast = \{\lambda_1,\dotsc,\lambda_e\}$ of $E^\ast$, defined by
$$\lambda_i(v_j) = \delta_{ij} = \begin{cases}1 &, i = j\\ 0 &, i \neq j. \end{cases}$$
It is then easy to see that $F^0 = \operatorname{span} \{\lambda_{f+1},\dotsc,\lambda_e\}$, whence $\dim F^0 = e - f = \dim E - \dim F$.
Best Answer
As $\phi: V \times V \rightarrow \mathbb{R}$ is bilinear, we can form a map from $V$ to its dual space $V^{\ast}$ by just putting $v$ into its first slot, so
$$V \rightarrow V^{\ast},\ v \mapsto \phi(v,-),\quad \text{where }\phi(v,-): V \rightarrow \mathbb{R}.$$
(I'm guessing that the orthogonality refers to the form $\phi$?) The orthogonal subspace to $v$ is $v^{T} = \ker(\phi(v,-)) = \{w \in V\, : \, \phi(v,w) = 0\}$. Viewing $\phi(v,-)$ as a linear map from an $n$-dimensional vector space $V$ to a $1$-dimensional vector space $\mathbb{R}$, the Rank-Nullity Theorem gives
$$ \text{rank}(\phi(v,-)) + \text{nullity}(\phi(v,-)) = \dim V \iff \text{nullity}(\phi(v,-)) = n - \text{rank}(\phi(v,-)).$$
Noting that the nullity here is just the dimension of the kernel, which moreover is the dimension of $v^{T}$, one gets that $\dim(v^{T}) = n$ if the rank is zero (i.e. $\phi$ is degenerate), and $\dim(v^{T}) = n-1$ if the rank is $1$ (i.e. $\phi$ is non-degenerate, so $\phi(v,-)$ has a trivial kernel).