In the Multiple Regression Problem With $k$ Explanatory Variable, It Would Be Quite Tedious to Derive the Would

In the multiple regression problem with $k$ explanatory variable, it would be quite tedious to derive the formulas for the slope coefficients without knowledge of linear algebra. The formulas certainly do not resemble the formula for the slope coefficient in the simple linear regression model with a single explanatory variable. However, it can be shown that the following three step procedure results in the same formula for slope coefficient of the first explanatory variable, $X _ { 1 }$ :
Step 1: regress $Y$ on a constant and all other explanatory variables other than $X _ { 1 }$ , and calculate the residual (Res1).
Step 2: regress $X _ { 1 }$ on a constant and all other explanatory variables, and calculate the residual (Res2).
Step 3: regress Res1 on a constant and Res2. Can you give an intuitive explanation to this procedure?