How exactly does money supply causes inflation? I think there's consensus among economists that a growth in money supply (I'm not thinking in any particular type of money supply) causes inflation (CPI), at least, in the long run.
What I'm interested in is the exact mechanism for that phenomenon. How "printing more bills" makes the price of, say, milk in the local supermarket raise? How does the people who set prices know that there's more money in circulation than before so it's time for a raise?

Thank you in advance.
The view that money growth causes inflation may have been a consensus view during the (old) Monetarist era, but it is unclear whether that still holds. 

(One somewhat related question here discusses whether inflation can occur without money supply growth - link)

As seen in the data around Quantitative Easing, we certainly see deviations between money supply growth and inflation "in the short term."

In the "long run" (which is what you asked about), it is unclear what causes what. Other than in Monetarist models (which are arguably no longer consensus), the money supply is determined by a portfolio allocation by the household sector. That is, people hold money instead of interest-bearing bonds for whatever reasons. What policymakers control is the policy rate of interest. Therefore, money growth is an outcome of the policy environment, like the inflation rate. In other words, faster nominal GDP growth (and hence inflation) tends to cause faster money supply growth.

However, if we want to assume that policy makers are controlling the money supply, and that money growth indeed causes inflation, modern theories would probably lean on the following two mechanisms.


Money does not just appear; it is a government liability, and the growth of government liabilities implies fiscal deficits. That is, loose fiscal policy raises demand, and causes inflation.
Expectations. If people are convinced that money supply growth causes inflation, they will watch the published money supply numbers. (In the United States, money numbers are published weekly, and were followed closely during the early 1980s. Currently, I doubt that 10% of bond traders under the age of 30 know what day of the week money numbers are published on.) They will raise prices based on what they think the money supply is doing now, and what they forecast it will be doing. I am not expert on Market Monetarism (the modern offshoot of "old Monetarism") but I believe that they emphasise this effect; although they might phrase things differently.