An ablative absolute is made up of a noun/ pronoun and a participle, agreeing with one another in the ablative case. We call these expressions 'absolute' because they have no grammatical link to the rest of the sentence and are free-standing. You can spot them by looking out for particples with ablative endings (Past/ Future- o, a, is. Present- nti, ntibus), agreeing with a noun in the ablative case. Ablative absolutes most commonly occur with a perfect passive participle, which can be literally translated as 'having been ..ed' (e.g. scriptus, a, um = having been written).
An example in English would be, 'after cooking dinner, the girl went out'. The main clause in this sentence is, 'the girl went out', and the sub clause, 'after cooking dinner', could be expressed with an ablative absolute since there are no grammatical links between the two. First we need to turn the English into a passive form - 'with the dinner having been cooked'. In Latin this would be, 'cenā coctā' (note the agreeing ablative endings). All together the sentence would be, 'cenā coctā puella exivit'.
Now from Latin into English: 'Caesar exercitu exposito, castra posuit'
Find the ablative noun and agreeing participle- 'exercitu exposito' = 'with the army having been landed.'
Tranlsate the main clause, 'Caesar... castra posuit'= 'Caesar made camp.'
Put the clauses together to figure out the literal translation = 'With the army having been landed, Caesar made camp.'
Rephrase the sentence so that it makes sense in English (this most often means subsituting the passive meaning for an active one, e.g. 'the army having been landed' becomes 'he landed the army) = ' After he landed the army, Caesar made camp'