Pressure emerges from collisions at the molecular scale
Pressure is the force exerted by gas molecules bouncing off a container wall. Each molecule has some velocity component perpendicular to the wall. When it collides elastically and bounces back, it transfers momentum to the wall (by Newton's third law). In a large container, trillions of molecules collide per second. The average force per unit area is pressure. Kinetic theory relates this to the average kinetic energy per molecule: P = (1/3) * n * m * <v^2>, where n is the number density, m is the molecular mass, and <v^2> is the mean square velocity. Hotter gas means faster molecules and higher average <v^2>, so pressure increases. Higher density (more molecules in the same volume) also increases collision rate and pressure. This microscopic picture explains why ideal gas law works: PV = Nk_B T is essentially the statement that pressure scales with particle density and temperature.
Molecular velocities and diffusion
The average speed of a gas molecule is not the same as the average velocity (which is zero). The root-mean-square speed is sqrt(3k_B T/m). At room temperature, nitrogen molecules move around 500 m/s on average. At higher temperatures, speeds increase. Faster moving molecules lead to faster diffusion: perfume spreads faster at high temperature because the molecules have higher kinetic energy. Effusion (gas escaping through a small hole) also depends on molecular speed. Lighter molecules (H2) effuse faster than heavier ones (CO2) at the same temperature, which is why hydrogen balloons leak faster than air-filled balloons.