Any equation in the form y=mx+b with a slope of 4 must have m=4 and to have a y intercept of -3, b=-3. You can put that together to make y=4x-3.
The reason the y intercept is -3 when b=-3 is because y intercept signifies the value of y when x=0. And when x=0, mx=0, so all you have left is y=b. Slope is just how quickly y increases compared to x.