It turns out, the saying originates in the United Kingdom where it can be pretty soggy in the spring.
In the United Kingdom and Ireland, an April shower is rain during the month of April. One of the major causes of the often heavy downpours is the position of the jet stream. In early spring, the jet stream starts to move northwards, allowing large depressions to bring strong winds and rain in from the Atlantic. In one day the weather can change from springtime sunshine to winter sleet and snow. The track of these depressions can often be across Ireland and Scotland bringing bands of rain followed by heavy showers (often of hail or snow) and strong blustery winds.en.wikipedia.org/wiki/April_shower
April Showers Bring May Flowers by Karen Chappell April showers bring May flowers, That is what they say. But if all the showers turned to flowers, We’d have quite a colourful day! There’d be bluebells and cockleshells, Tulips red and green, Daffodils and Chinese squill, The brightest you’ve ever seen. You’d see tiger lilies and water lilies, Carnations pink and blue, Forget-me-not and small sundrop Glistening with the dew. We’d have fireweed and milkweed And many more different flowers. Mexican star and shooting star, Falling in the showers. And if all the showers turned to flowers On that rainy April day, Would all the flowers turn to showers In the sunny month of May?