Why One Side of Mountain Range is Lush
The Cascade Mountain Range in the US Pacific Northwest is a good example to use to explain this. The predominant wind direction is from the West - over the Pacific Ocean. The air over the ocean picks up moisture from evaporation. After it passes the coast, the mountains cause the air to rise. As it does so it cools. Colder air can hold less water vapour than warm air so that the air becomes saturated with water and it condenses out into clouds. With enough condensation you get rain - a lot of it on the western side of the mountains. So you had it backwards, the rain doesn't allow the air to rise - the rising allows it to rain. Clouds mainly approach from the lush side because that is the prevailing wind direction but it is better say that it is the lush side because clouds mainly approach that way. When wind comes from the other direction is travelling across land so it contains less water and the cooling generally isn't enough to cause rain.

Once the air crosses to the other side of the mountain range it warms and becomes undersaturated with water. Clouds often disappear and even if they remain you aren't getting further condensation to produce a lot of rain. So those areas are much drier.