The double spending is solved by the presence of a central server. Once you send the token to someone, that someone shall query the mint to understand if the token has been already spent or not. If so, then the token is invalid. If the token is not present in the mint database of spent coins, then the token is valid, the mint collects the token and returns to you a new token of the same value.

Actually a "token" is a bunch of tokens, like $100 in cash can be made up by some banknotes. A cashu ecash token can be of value 2^n, thus 1, 2, 4, 8, 16, ....