As this paper is about a Curry-Howard correspondence, the statement has two (equivalent) readings.

Reading 1. It's a typing judgement. You can read ∆⊢e:A as "the program e has type A, assuming the free variables in e are typed as described by ∆".

Reading 2. It's a logical judgement. You can read ∆⊢e:A as "the proof e is a proof of the logical formula A, assuming the free variables in the proof e stand in for proofs as described by ∆".