洗洗 is the reduplicated form (VV) of 洗 'to wash'. In Chinese, repeating a verb one time (VV pattern) makes the action sound brief, casual, or softer — you can think of it as 'to do a little washing'. This pattern is especially common in suggestions and polite requests (你洗洗吧 'why don't you do a bit of washing'). Other common examples include 看看 (take a look) and 听听 (listen for a bit).