Galileos statement about object thrown in horizontal direction arriving on the floor at the same time as if free fall

The statement is correct as long as the ground is horizontal and aerodynamic forces such as drag or lift can be ignored (so it doesn't apply when throwing a frisbee, for example). The effect of gravity on a bullet's path is known as bullet drop.

To understand this, it is necessary to understand the concept of velocity and acceleration as vectors. There are plenty of excellent free resources online to explain you this.

https://youtu.be/q9IWoQ199_o

https://youtu.be/0na1JdPE_JY

https://youtu.be/k6aJyOHTDYM

Basically what happens is that both the velocity and acceleration vectors along the y - axis (vertical direction) are same in both cases. Hence time to fall through the same distance in y direction is same.

Let's do an experiment!!

Hold one end of a string whose other end is tied to a stone. When you pull the string up (apply force in vertical direction), the stone just moves up with the string. However, now hold the end of string still and push the stone in horizontal direction and let it swing (this system is called simple pendulum) while the system is in motion pull up the string like before. This time the ball not just moves up as you pull the string up but also swing horizontally. In short, you cannot stop the horizontal swing of pendulum stone by applying force in vertical direction!

A force applied vertically cannot change the horizontal velocity of a moving body, the force just gives vertical velocity to the body without altering the horizontal velocity!

Similarly, earth applies the same force and in same direction on a bullet at rest and a bullet moving horizontally, the force has no effect on horizontal motion of bullet and neither does the horizontal velocity has any effect on force (obviously!) So, the vertical velocity changes in same way in both the horizontally moving bullet and the bullet at rest; and it takes the same time in falling down (Although the bullet must have travelled hundreds of metres horizontally before hitting the ground so it seems that it took longer time but that's not true!!)