Translated from Ustadz Firanda Andirja's Indonesian commentary (Bekal Islam). Quoted verses, hadith and duas keep their Arabic; where the commentary gave its own Indonesian rendering of them, that rendering is not translated and is marked [translation not shown].
The meaning of tasbih is declaring Allah free of every deficiency and flaw, and of every corrupt assumption about Allah. Linguistically, tasbih means to keep far away (Mu’jam maqooyyis al-Lughoh 3/125). So tasbih as a technical term means keeping the heart and mind far from the assumption that Allah has any deficiency, or that any evil is attributed to Allah. This includes keeping far from Allah / declaring Allah free of every deficiency that the polytheists and atheists accuse Him of.
At-tasbih (declaring Allah free of imperfection) entails at-ta’dziim (magnifying) Allah, and magnifying Allah entails affirming praises of Allah (see Majmuu’ Al-Fataawa 16/125)
So the meaning of سُبْحَانَ اللهِ وَبِحَمْدِهِ is: [translation not shown]