The reason why calendars in many countries (like the US and Canada) talk about June 21 as the start of summer is because it uses the so-called astronomical seasons, in which one goes from solstice to equinox or vice versa - e.g. summer being from the summer solstice (June 21 or so) to the autumnal equinox (Sept. 23 or so). read more
Most people consider the first day of summer to be the Summer Solstice, which rings in at 6:07 a.m. EDT on June 21 this year. This is because, astronomically, the sun will be directly overhead of the Tropic of Cancer at that time. read more
The "Summer Solstice" does not necessarily mark the "beginning of Summer". In fact, that is when daylight starts to shorten. You are talking about the astronomical seasons. There are the "Meteorological Seasons" which weather scientists study. Those start on the 1st (June 1 for Summer). read more