Why is horizontal velocity constant in projectile motion?

Horizontal velocity in remains constant because we neglect air resistance (aka drag) so there are no horizontal forces acting on the projectile, only the vertical force of the projectile's weight.

It is important to remember the definition of a projectile. It is an object that falls under the influence of gravity. Objects using onboard engines are not projectiles under this definition.

When you shoot a projectile at an angle to the horizontal, the vertical component of the velocity decreases in the upward direction because the weight causes a vertical deceleration. The vertical component will eventually reach zero. The projectile will then accelerate downwards. (In the absence of drag the trajectory is symmetrical.)

Now, while the projectile is flying through the air, there is no other significant force acting horizontally. The horizontal gravitational attractions of other bodies like large buildings and the Moon are not strong enough to make the horizontal velocity change observably.

Even though air resistance is neglected to enable the use of the equations of uniformly accelerated motion and constant velocity, results using these equations are nonetheless accurate to a good level of approximation. Those results are therefore informative.

The only exception to this is when the projectile is shot to a very far or high distance, then the rotation of the Earth causes the relative position of the projectile to change. At that point, you have to take rotation of the Earth into account, but usually you don't have to worry about that unless you are launching a rocket into space to a particular target.