On 22nd December, the northern end of the earth’s axis is inclined away from the sun. This day is known as the winter solstice. Thus one may assume that, in the northern hemisphere, June should be the hottest month of the year and December should be the coolest, because they are the months which receive the largest and least amounts of radiation from the sun. But, usually there is a lag because time is needed for the heating and cooling to become effective. For this reason the warmest months in the northern hemisphere are June, July and August, while the coldest months are December, January and February, that is, a little after the summer and winter solstices. June, July and August are thus important for the summer monsoons because the tropics are warmest at the time of the year.
But, it is important to realize that monsoonal winds depend on the thermal contrast between the continents and oceans. The period when the continents are warmest need not necessarily represent maximum thermal contrasts, because the oceans respond differently to solar radiation.
Acknowledgement: Images from wikipedia