Why does is continue to warmer or colder after the respective summer and winter solstice?

Because there's very little change in solar declination for at least a month after each solstice, so heat either continues to build up, or cool down. By the time of the summer solstice, the sun's declination has been 20 degrees plus for only a month. By July 21, it's been at 20 degrees plus for TWO months. Since the sun has been essentially just as high for an extra month, July 21 is bound to be hotter on average than June 21. The reverse is true for January 21. By then the sun has been 20 degrees or more below the celestial equator for a month longer than on December 21, so it's bound to be colder on average.