Popper starts out this short excerpt by explaining (usefully) what the Utopian Method is. The method stipulates that to judge an action as rational, it must first be established what the aim is and then determine the best means to achieve this end. The Utopianist can only judge an action as rationally with an established end and only relative to that end.
The method is applied to politics in such a way that e.g. a politicians aim it is to restructure the state. To do so, the politician need to define, as precisely as possible, what a good state is and can only then determine the means to slowly drift towards that end.
This is Utopianism and, Popper quickly follows, this idea is as attractive as it is pernicious.
Ends can neither be determined scientifically nor by pure reasoning, making Utopianist approaches self-defeating. The Utopianist can present arguments for her view and those who will listen might be swayed but she cannot change those views which are not open to change. To explain this concept better, Popper uses a great example: the Physicist.
No amount of physics presented to a physicist will compel him to decide that it is the right thing to build a bomb or an airplane or a plough. To make such a decision the physicist has to either adopt or be given an aim. Consequently, the scientists will only construct means to an end.
Popper’s critical point now becomes clear: he worries that the Utopian method will, or must lead to violence.
The Utopian method can only work if all agree to and end. If the Utopianist is not among like-minded people or has the power to present an argument accepted by all, she has no choice but to destroy opposing Utopian views.
Thus, to judge an action solely on an end which cannot be determined by rational or scientific means will inevitably lead to the need of the Utopianist to resort to violence.