When you fit a linear model, the statistical softwares give you the point estimate, confidence interval, test statistics, and p-values of the $\beta$_s. If you are just interesting in these $\beta$s, you can stop here (for example, the simple linear regression just have on intercept and one slope, so $\beta$s themselves are enough). But for little complicated model, you will not satisfied by $\beta$s, and you want to estimate, test the linear combinations of $\beta$s. At this time The importance of C matrix is obvious. For complicated model, such as model with interactions, C matrix must needs be constructed.
Example 1: For ANOVA, one categorical covariate has 3 level. Suppose level 1 is reference. Two $\beta$ will give you the difference between level 2 vs 1 and level 3 vs 1. If you want the difference between level 2 and 3, you need C matrix (0 1 -1). (the first 0 is for intercept). If you want to estimate the means for level 3, C=(1 0 1) is needed.
Example 2: If you want to the multiple hypothesis simultaneously, see Testing the general linear hypothesis: $H_0: \beta_1 = \beta_2 = \beta_3 = \beta_4 = \beta$. Here T=C.
Example 3: If the interactions exist, we need to have the linear relation for each combination (cell) of interaction. Here is 16x16 C matrix to get 8 intercepts and slopes. How to understand the coefficients of a three-way interaction in a regression?
You can find more example on the internet, textbooks.
In summary, for linear model, constructing C matrix is equal to half of theory of linear model.