When logging in to your LogMeIn account, you select "I trust this device, keep me logged in" but your computer is not saved as a trusted device and you are prompted to input your login credentials the next time you try to log in from the same device.
In LogMeIn Central, if Two-factor authentication
is enabled under Users
> Login Policy
by the Master Account Holder, the Login Policy does not allow you to save devices.
Contact your LogMeIn account's Master Account Holder and request that they change the Login process