As the comments have said, the second and third numbers are the ratio of sidereal to solar days but it is split into two parts from the original formula which gives GMST at 0h UT1 by including the hours part directly.
The first number comes from the definition of sidereal time as "in general terms, sidereal time is the hour angle of the vernal equinox" (Explanatory Supplement to the Astronomical Almanac, 1992, p48 (referred to on p77 of the 3rd Edition of the Explanatory Supplement)). This is bound up with the original definition of the right ascension of the fictitious mean sun which moves around the celestial equator at a fixed rate which was due to Simon Newcomb in 1895 (Astronomical Papers Prepared For The Use Of The American Ephemeris And Nautical Almanac Volume 6 (1898). This was based on noon and shifting it by 12 hours to the definition used now at converts the 18.64606 to 6.646065. Differences in the measured values of the precession constants and changes in the star catalog reference systems from FK4 to FK5 have resulted in changes in the values from the original.
Aoki et al. 1982 discuss this in more detail and added a $T^3$ term (their equation 13, although note the values there are in seconds not hours as in the original USNO version above) for what became the IAU 1984 system. Capitaine et al 2000 added a $T^4$ term along with another 32 more terms to the so-called equation of the equinoxes which is added onto the formula for GMST to get GAST (Apparent Sidereal Time). The equation of the equinoxes takes account of the accumulated precession and nutation as the fictitious mean sun is not a very good reflection of reality for precise work. Because of these increased complexities and coupling between the Earth's rotation and the background reference frame of the stars needed for precise work, the IAU did away with sidereal time in 2006 and replaced it with Earth Rotation Angle (ERA) which is a simple linear function of UT1 time.