Answer: Because mosquitoes breed and emerge better on humid conditions. Moreover, mosquitoes out of range 24-27 degrees Celsius become less active with shorter lifespan.
Step-by-step explanation:
Honestly, mosquitoes are there with us in basically every season and time. It's usually seen in mid-late summer/early fall because lots of country have rainy season in that season(what matters more is being wet than being warm).
They can live anywhere as long as they have stagnant water for enough time to have success in the larval period.
This is supported by several evidences:
It breeds not only in temperate/tropical regions, but also in cold places such as Siberia and the Arctic. Although, these mosquitoes in specific has evolved to fit their environment and are very vulnerable to the heat of the temperate/tropical regions. It sleeps in eggs during winter and spring, spending time for the freezing temperature of minus 50 degrees Celsius, and simultaneously hatches in summer when the weather is warmer.
Especially in Siberia and the Arctic, very short summers become mosquito hells, making it difficult for warm-blooded animals in the region to even open their eyes for a slight second to the attacks by blood dwelling mosquitoes. Reindeer of this period for instance, cannot withstand the crazy mosquitoes' blood sucking, and about 1% of them die of mosquitoes. Still, mosquito activity and reproduction rates are less than temperate/tropical climates because it is only possible to be active on short period of warm, summer days in cold climates like this.
But still, this means that mosquitoes can survive cold temperatures over winter, but chooses to be active in summer mostly.
So back to our question,
Yes, the temperature matters.
But the factor that mostly affects the mosquito population is not temperature, but humidity. When it rains, the climate becomes humid, breeding and growth are supported, and the population increases dramatically.
So if it rains often, you'd better prepare for mosquito strikes.
Mosquitoes basically lay eggs in water, so when there is less water available, its population decreases. In fact, even in the same area where mosquitoes are found(although there is literally nowhere in planet earth where mosquitoes are not found,) if a drought is severe, the population decreases sharply. In addition, when heavy rain occurs frequently, such as typhoons, the population can also be greatly reduced because all the mosquito larvas in the puddles and stagnant water are swept away.
For areas where four seasons are clear, the population of mosquitoes would mostly increases in autumn when the climate becomes relatively reasonable. Mosquitoes are most active at 24-27 degrees Celsius because they become less active with shorter lifespan when the heat or cold is excessive (32°C or higher), so they choose to be inactive out of that range.
To sum up, mosquitoes can live in anywhere, at any place.
But they are most active in warm temperature, with plenty of water sources, and this is why mosquitoes are seen most often during summer/rainy season.
In addition,
Nowadays, there's a whole different story for mosquitoes. Unfortunately, human beings have similar optimal temperature range with mosquitoes, and we run air conditioners when hot, and heaters when cold. This makes excellent condition for mosquitoes to breed and reproduce inside a building, although there must be a stagnant water source left alone for enough time.
Hope this helps!