Why does an economic slowdown lead to deflation? Usually economists say that in recession there is deflation, so increasing the money supply does not lead to a high level of inflation.
According to the Quantity theory of money, the price level is defined as such:
P = M*V/Y

Which implies that if the economy slows down, the V (velocity of money in circulation) goes down as well, leading to lower price level, hence lower inflation.
But what I don't understand, that at the same time the Y (Real GDP) goes down as well, cause of the recession, which should lead to a higher price level, hence higher inflation.
First, let me address some incorrect premises in your question

Usually economists say that in recession there is deflation, so increasing the money supply does not lead to high level of inflation.

This statement is not really correct. First, I dont know many economists who would say that usually recessions are deflationary. For example, according to Romer's Advanced Macroeconomics 4th ed, the macroeconomic aggregates in 11 recessions between 1947:1 and 2009:3 can be summarized as follows:

So out of 11 recessions during the time period under considerations only 5 were characterized by deflation, and average change in inflation over all 11 recessions was only $-0.3\%$ which indicates that the recessions with deflation had on average slightly higher effect than the recession with inflation, but the datapoint is so close to zero that it would most likely be statistically insignificant. So the statement you say about most economists saying recessions are deflationary is simply not factually correct. Or at least if there are economists saying that they definitely cannot be in majority and assuming that statement was correctly representing their views they would not be factually correct.
In fact as you can read in this old Economics.SE question in economics we even distinguish between inflationary and deflationary recessions acknowledging both types of recessions exist. Generally speaking you will find that supply driven recessions are more on the inflationary side (good example are the 1970s energy crisis) and demand driven recessions on deflationary side (great example of deflationary recession is Great Depression or Great Recession both demand driven recession).
Second, the two effects are not mutually exclusive with each other. For example, it can be that velocity of money drops by $10\%$ while real output drops only by $8\%$ leading to deflation. If output would drop by more than velocity you would get inflation. Money supply also matters as it definitely does not stay constant during recession. For example, Fed with its dual mandate usually tries to expands money supply, directly or indirectly, during recessions to stimulate the economy. In addition as disused by Krugman (1998), what matters is not just the actual change in money supply or other quantities but what are peoples expectations about changes in money supply and so on. An increase of money supply that is not credible to be permanent and expected to be quickly reversed will have no effect on inflation as if it would never even happen.
One important thing to note here is that when nominal interest rates are at zero lower bound (ZLB) it is correct to say that even large monetary expansion wont necessary be inflationary. The reason for that is that if you would want to go significantly below ZLB the peoples will strictly prefer to hold cash and any increase in money supply will be completely offset by drop in velocity of money. In such cases indeed increasing money supply wont lead to higher level of inflation but that certainly does not apply to any or as data above show even most of the recessions.