Understanding Laplace Transform equation – Mathematics Stack Exchange

I am currently studying mathematics in Digital Control Systems, and have now stumbled upon Laplace Transform. Could someone explain to me why this equation is written as it is?

Laplace Transform Equation

I understand this is supposed to be a “differential equation”, however, why do we have an exponential function raised to the power of “-st”, and what is that letter to the far left and why is our f(t) in square brackets as well as why it is suddenly changed to big F(s)?