The intercept (sometimes referred to as the “constant”) shows up in a regression formula. The intercept is the expected average value of the response variable (practicing lawyers in a state) if all the predictor values were zero. If none equals zero, then the intercept has no meaning and tells nothing about the relationship between the predictors and the response. With our two-predictor model, using only companies with fewer than 500 employees and total enrollment in top-100 law schools, the intercept is -1,917, which absurdly says that if all the states had no top-law-schools and no small companies, the model’s estimate for the number of practicing lawyers would have MINUS 1,917 lawyers. But of course, both predictors are not zero.
When no predictors equal zero you have a reason to center them. That means you re-scale them so that their averages do equal zero (software subtracts the average of the predictor’s values from each value). Now the intercept has meaning. It’s the average value of the estimated response variables at the average of the predictor variables. Returning to our model, when the two predictor variables are centered, the new intercept estimates 13,612 lawyers: precisely the actual average of the practicing lawyers in all of the states.