The estimated coefficients in multiple regression are "slopes" of the linear relationships, when other explanatory variables are held constant.
When you are dealing with multiple linear regression with variables $X_1, ..., X_m$, the estimated linear relationship between a variable $X_i$ and the response variable $Y$ implicitly holds all the other explanatory variables constant. This represents the "partial" relationship between the variables. This is different from the "marginal" relationship you will see when to create a scatter-plot of $X_i$ and $Y$, where the other explanatory variables are allowed to vary freely. As a result, the slope evident in the scatter-plot will not be the same as the estimated linear coefficient corresponding to that explanatory variable. They may be very different, and even have different signs.
To represent the relationship between $X_i$ and $Y$ while holding the other explanatory variables constant, you should use an added variable plot (also called a partial variable plot). On the horizontal axis you put the residuals of regressing $X_i$ against the other explanatory variables, and on the vertical axis you put the residuals of regressing $Y_i$ against the other explanatory variables (not including $X_i$). This scatter-plot will have a slope that is equal to the estimated linear coeffifient from your multiple regression.