The thing is, temperature is just motion at very small scales. On a solid, particles are attached to one another via the atomic forces, and a "temperature increase" means that the particles are jiggling more. With high enough jiggling, the particles start to break apart and the substance undergoes a phase transition. Now, when it's a gas, the particles are jiggling so much they aren't attached to one another, they are flying around at very high speeds as shown in the visualization. So, your intuition is correct: increasing the speed of the particles then is in fact the same as "increasing the temperature" of the gas.