In this problem you have what is essentially paired-data, with an equal number of observations of features from two objects/classes, where values in the classes can be correlated with one another (i.e., they are not assumed to be statistically dependent). Because correlation between features can occur, the overall variance of any linear combination of the feature values will depend on their correlation. This means that we need to know the variances of the individual features, but also the covariance between them. This is the reason we use a whole covariance matrix in this case.
In this case, each random vector $\mathbf{X}_i$ has some mean vector $\boldsymbol{\mu}_i$ and some covariance matrix $\mathbf{C}_i$. The latter is a full specification of the variances of each feature in the random vector, and also the covariances between the features in the random vector (from which we get the correlations of the features). A "linear discriminant" is effectively just a measure that aggregates the features by some linear function, so that we can compare an aggregate measure across multiple objects/classes. If we use the linear discriminant $Z(\mathbf{X}_i) = \boldsymbol{\beta}^\text{T} \mathbf{X}_i = \sum_k \beta_k X_{i,k}$ then the mean and variance of this "discriminant" can be found using the ordinary rules of mean and variance of linear functions:
$$\begin{equation} \begin{aligned}
\mathbb{E}(Z(\mathbf{X}_i)) &= \mathbb{E}(\boldsymbol{\beta}^\text{T} \mathbf{X}_i) = \boldsymbol{\beta}^\text{T} \mathbb{E}(\mathbf{X}_i) = \boldsymbol{\beta}^\text{T} \boldsymbol{\mu}_i = \sum_k \beta_k \mu_k, \\[8pt]
\mathbb{V}(Z(\mathbf{X}_i)) &= \mathbb{V}(\boldsymbol{\beta}^\text{T} \mathbf{X}_i) = \boldsymbol{\beta}^\text{T} \mathbb{V}(\mathbf{X}_i) \boldsymbol{\beta} = \boldsymbol{\beta}^\text{T} \mathbf{C}_i \boldsymbol{\beta} = \sum_k \sum_l \beta_k \beta_l C_{i,k,l}.
\end{aligned} \end{equation}$$
From these formulae we see that the variance of the discriminant $Z$ depends on the covariance matrix $\mathbf{C}_i$ and is a quadratic form. This is merely a result of the fact that when individual features are correlated, the overall variance of the "linear discriminant" is affected by these correlations. If you would like to learn more about this, I would suggest learning a bit about probability results for random vectors, and in particular, look at moment rules for linear functions. The variance of a linear combination of random variables is a quadratic form, as in the above formula. Hopefully this explanation gives you a starting point for this inquiry.