How do I close my late mother’s checking and savings account closed and get the small aount of money that she had in them?

I am so sorry for your loss.  Every state - and Missouri is not an exception - has a proceeding for a small estate.  In your state you would do a Small Estate Affidavit and post a bond for the amount left in the estate in case there were outstanding debts.  The law requires that you assume responsibility for money owed to creditors.  Here is the law:

RSMo 473.097 was created by the Missouri Legislature so that heirs of a small estate had an avenue to more efficiently administer the estate. The specific statutory language says that much of the probate process can be circumvented if:

(1) The value of the entire estate, less liens, debt, and encumbrances, does not exceed forty thousand dollars;

(2) Thirty days have elapsed since the death of the decedent and no application for letters or for administration or for refusal of letters under section 473.090 is pending or has been granted, or if such refusal has been granted and subsequently revoked;

(3) A bond, in an amount not less than the value of the personal property, approved by the judge or clerk of the probate division is filed by the person making the required affidavit conditioned upon the payment of the debts of the decedent, including any debts to the state of Missouri, the expenses of funeral and burial and compliance with future orders of the court in relation to the estate of the decedent; and further conditioned that any part of the property to which the distributee is not entitled will be delivered to the persons entitled to the property under the law. Liability of the sureties on the bonds provided for in this section terminates unless proceedings against them are instituted within two years after the bond is filed; except that, the court may dispense with the filing of a bond if it finds that the same is not necessary;

(4) A fee [...], and when required, the publication cost of the notice to creditors are paid or the proof of payment for such publication is provided to the clerk of the probate division.

Good luck!