The original sentence you wrote is mostly correct but can be made a little clearer. Here is your sentence with a slight improvement:
When spring arrives and the snow melts, male capercaillies put on a big show. They fan their tails and make funny coughing and popping noises to get attention.
This sentence is clear and uses proper grammar. The verb tenses are consistent, and the commas are correctly placed after the introductory phrase. The words are easy to understand for a 6-year-old too.
So, great job! If you want a simpler version for a young child, you could say: "When spring comes and the snow melts, boy capercaillies show off. They spread their tails and make funny sounds to get noticed."