Literally translated, the sentence would be "in America, when becomes spring, rain often falls", or more naturally "there is regular rainfall in America during the springtime".
で, in this case, simply marks the location that the action described by the verb (降る) takes place. Here, it represents the English 'in', but it could also mean 'at', such as in 「クラブでかわいい女の子に出会ったよ」"I met a cute girl at the club", or 'with' such as in 「傘で刺されちゃった」"I was stabbed with an umbrella", so it's best not to associate it with an English word, but rather accustom yourself to it through familiarity.
As I'm sure you're aware, は marks the topic of discussion (for an in-depth explanation of は and the differences between は and が, among other things, check out the excellent book 'Making Sense of Japanese' by Jay Rubin). Since America is both the topic and the location of the action it takes both particles では.
と sets up a conditional, which in Japanese can mean either 'if' or 'when' depending on context, and can be done by use of と, ~ば, or ~たら. They are essentially translated the same in English, but the nuances differ in Japanese. と can't be used for suggestions or happenstance, it's more definite so it describes factual conditions or habitual events, so you can see why it is used for describing seasonal changes.
Hopefully this is helpful; posts like this are a good means of review for me at least