why does a hot-air ballon rise in the sky?!!

A hot air balloon for manned flight uses a single layered, fabric gas bag (lifting "envelope"), with an opening at the bottom called the mouth. Attached to the envelope is a basket, or gondola, for carrying the passengers. Mounted above the basket and centered in the mouth is the "burner" which injects a flame into the envelope, heating the air within.

****** Raising the air temperature inside the envelope makes it lighter than the surrounding (ambient) air. This causes the balloon and its payload to rise. ******

{Note1}
(lighter than air) - means less dense than air. Density (mass per unit volume) is the critical property, not mass and not weight.

{Note 2}
(Hot air) - The density of a gas can be reduced by raising its temperature while leaving the pressure unchanged (Boyle's law).

{Note 3)}
(Boyle's law) - Boyle's Law states that the product of the volume and pressure of a fixed quantity of an ideal gas is constant, given constant temperature. Expressed mathematically, the formula for Boyle's law is:
is:
p V = k
where:V is volume of the gas.
p is the pressure of the gas.
k is a constant (see Note 1).

The value of k is computed from measurements of volume and pressure for a fixed quantity of gas. After making a change to the system, typically by forcing a change in the volume of the vessel containing the fixed quantity of gas, the new volume and new pressure are measured. The result of computing the product of the measured new volume and the new pressure should be the original value of the constant k. Without being too rigorous at this point, the equation says that, after forcing the volume V of the fixed quantity of gas to increase, keeping the gas at the initially measured temperature, the pressure p must decrease proportionally. Conversely, increasing the volume of the gas decreases the pressure.