Push vs weight. A rocket rises only if the engine push (thrust F) is bigger than its weight mยทg. The ratio F/(mยทg) is the thrust-to-weight ratio. Below 1 it stays on the pad, however much fuel it carries.
The rocket equation (Tsiolkovsky). Fuel is thrown backwards, so the rocket is pushed forwards. The speed gained in empty space is
ฮv = Isp ยท gโ ยท ln(m_start / m_end)
Isp is the engine's efficiency in seconds, gโ = 9.80665 m/sยฒ. The logarithm means each extra bit of fuel helps less than the one before, and dropping empty tanks (staging) makes the rest lighter, so the top stage gets a better ratio.
Gravity. g = GM/rยฒ gets weaker with height. The escape speed is v = โ(2GM/r) (about 11.2 km/s at the surface). The Moon is about 384,400 km away; a rocket that coasts with a little less than escape speed would fall back, so reaching the Moon needs roughly the same energy as leaving Earth.
Gravity loss. While the engine burns, gravity keeps pulling, so the speed you really get is smaller than the rocket equation says. The chart and the experiment show the gap.
Independent check. The page integrates F = mยทa with a changing mass in empty space and compares it with the formula:
โฆ
Simplifications (honest list). The rocket goes straight up (real rockets turn sideways and circle the Earth first); no air drag; the Moon's own pull and the Earth's spin are ignored; air only changes the engine Isp between sea-level and vacuum values; tanks and engines are invented, very light "toy" parts, so real masses are far larger; reaching 340,000 km counts as "reaching the Moon"; the Moon and Earth in the picture are not to scale. Fins only keep real rockets steady inside air, here they just add weight. Not any real rocket, company or agency.