What are my options if the manager of my apartment complex wanted me to give up my lease for the stains in my carpet?

I do not understand, if the stains are in your apartment's carpet, why that would be of current concern to the manager. If they are your fault it is possible that you will be liable for their removal or a replacement carpet.

Your security deposit covers damages such as these, although you can be personally responsible if the damage exceeds the deposit. If you are month to month she can give you 60 days notice and there is nothing you can do about it (in most jurisdictions). If you have a term lease that has time on it she can only force you to leave if in creating the stains you somehow violated the lease, i.e. cooking meth, freebasing cocaine, that type of thing.

Tell her no, you will not be leaving. You are complying with the lease and so must she.