What is the meaning of "I do that for the sake of Allah," and how can one's intention be purely for Allah in acts of worship such as prayer? Is what is meant by doing something for the sake of Allah that it is done in compliance with His command?

As for obsessive whispers (waswasah), do not pay them any mind. The matter of intention (niyyah) is straightforward. The meaning of sincerity (ikhlas) is to act for the sake of Allah, by performing acts of obedience desiring nothing but the countenance of Allah the Almighty. So, you pray seeking Allah's pleasure, not for people's praise or worldly gain.