The HP Filter has two objectives, with the importance of each objective denoted by the user given value of lambda:
Objective 1: minimize the $\tau_t$ in the term in the square brackets such that we minimize the changes in the estimated growth rate over time.
Objective 2: We want to bring the $\tau_t$ to be as close as possible to $y_t$ to minimize the first sum in the equation.
What I am failing to understand is why these acts of minimization will help us discover what the cyclical component of GDP is? Is there a fundamental gap in my knowledge?