Residual plots: why do we want to know the error?
Let me try to give an example: Let's say we would like to predict a housing price by looking at the number of bathrooms. The x-axis represents your feature/independent variable (#bathrooms) and your y-axis represents the response/dependent variable. The points represent training samples you collected and know both the independent and dependent variables of.

Now in Linear Regression our goal is to predict the housing price based on a new example of #bathrooms we didn't see yet. For that we have to create a line through the data points that represents the training data the best. So, what qualifies a line as being a good or bad line? The answer is a cost/loss function. In linear regression we often use the squared mean of the error/residuals (MSE):

$$
\frac{1}{n} \sum (y_i - \hat{y_i})^2
$$

where $y_i$ is the true housing price and $\hat{y_i}$ is the predicted value.
Our goal is to minimize this loss.

Plotting the residuals can give us many insights: For example on which training samples we make the highest errors. Also we can see how the residuals are distributed. For more complicated applications we need to check this distribution for the so called Markov Theorem which would go too much in detail.