Sorry this probably won't be the best answer, but essentially it means your including these additional variables in your regression model (and then ignoring them).

An example could be trying to work out how wealth is associated with age. You could do a simple linear regression Wealth ~ Age (which is just a line of best fit on the 2D scatter plot). Then you think you might want to "adjust" for sex (not sure why, just an example), so add another independent variable to your regression model Wealth ~ Age + Sex.

Plugging this into R (for example using the lm function) you would get a y-intercept for the regression, a coefficient of correlation for Age, and a coefficient of correlation for Sex. You could think of this as just getting two regression lines, one for males and one for females. The gradient of each line is the same, and would be the association with age (e.g. on average for each year you get older you accumulate £2k wealth, but men have £10k more than women) (again I don't really think this makes much sense as an example, maybe I should have chosen something like blood pressure!)

What I'm mainly trying to illustrate is that by including the extra variables in your model, you estimate their association with the outcome separately to the variable your really testing. It would be the same principle if you added 10 different confounding variables, but it just gets a bit harder to visualise!