I've recently started learning about data science/statistics and learned how to derive such models as linear regressors and logistic regressors. What I don't understand, however, is what the parameters that are calculated in a logistic regression correspond to. For instance, it's simple in linear regressors in which $X \in \mathbb{R}^{2\times n}$ with $n$ training examples and $y\in\mathbb{R}^{1\times n}$. So long as the $X_0$ (the first column of X) is all ones and the second column correspond to the X coordinates and y contains the y coordinates of the examples, the calculated parameters in linear regression are in the shape of $\theta\in\mathbb{R}^{1\times 2}$. Clearly, $\theta_0 = b, \theta_1 = m$ as the regression on a linear equation $y=mx+b$.
Furthermore, the parameterization of linear regression makes clear sense to me because the calculated parameters in $\theta$ directly correspond to coefficients in the line function. However, I have a lot of trouble understanding what the parameters calculated by multinomial logistic regression correspond to. I understand that logistic regression is used for classification and that the parameters calculated must in some way distinguish between the classes based on the input data, but I don't quite understand how. A graphical explanation would be preferred if possible :)
Thanks in advance!