All pressure$^1$ is due to a material wanting to expand.
Pressure is defined as a force over an area. In the case of a gas, the force isn't a constant but comes from the time and area average of many collisions with your measuring surface. You can increase the average force by either increasing the number of particles colliding each second by increasing the density$^2$, or you can increase how hard each particle hits by increasing their velocities by increasing the temperature. You can see this from a version of the ideal gas law:
Where $P$ is pressure, $\rho$ is the density, $T$ is the temperature (in absolute units)$^3$, and $R^*$ is the specific gas constant.
In the case of a balloon the temperature should be roughly equal on each side of the balloon and isn't much in our control$^4$. Thus to fill a balloon we must cram more and more molecules into the balloon so the density is high enough to balance the atmospheric pressure.
So then one may wonder how people can blow up a balloon with their lungs, if the weight of the atmosphere is creating the pressure on the outside of the balloon. The answer is that the atmosphere helps us out by compressing the outside of our lungs helping us push the air into the balloon.
$^1$ Dynamic pressure also includes pressure due to a change in velocity, like a wave pushing someone over. However, static pressures are all about the material trying to return to its most relaxed state. Sometimes, in the case of liquids and solids this can mean there's a negative pressure if the liquid or solid is being stretched.
$^2$ The number of collisions is also increased by increasing the velocity of each collision. This means that the velocity gets to contribute twice. Fortunately temperature is dependent on the average square velocity, so the pressure just depends linearly on temperature.
$^3$ The temperature must be in absolute units because the pressure would only be zero at absolute zero, not say the freezing point of water (Celsius), or a saltwater slushy (Fahrenheit).
$^4$ If you place the balloon in the freezer, its temperature will decrease, but the atmospheric pressure will still be pushing on the outside so the balloon will collapse until the pressure is balanced by the increased density.