Increase the price of labour and the employer's demand for it will fall. Sounds plausible doesn't it? Simple supply and demand economics surely?
Well no, actually, unfortunately for you libertarian neo conservatives, your model is far too simple (and I suspect at least some of you know this).
People are not loaves of bread. If you increase the price of a loaf of bread, it is still the same loaf of bread, so demand will obviously fall (except in giffen good cases etc.).
But increase the price of labour and wonderful things can happen; labour turnover falls, motivation of workers can increase, lazy managers can't get away with ignoring their low paid workforce as better trained and organised staff are needed to justify the higher salaries, and yes, more robotization. All these things increase productivity and are a good thing not just for the added efficiency but also for social cohesion as it reduces inequality.
But, you may ask, surely employers aren't stupid, they will train their workforce better and increase robotization anyway if it improves efficiency? The market will demand it surely?
Afraid not. Any employer that unilaterally decides to do this (unless they have massive reserves of capital and long term foresight which most companies don't have) will in the short term place themselves at significant disadvantage and added risk. Very few (if any) will do this even though it might be in their long term interest.
This is where governments come in to provide a regulatory framework for the market to work properly (all economic miracles UK, US, Japan etc. were based on government support - even initial protectionism, definitely NOT laissez-faire).
When a minimum wage is set, companies have to compete more on innovation of production/service, organisation and training of its staff, which is in the long term interests of all of us.