The book and its hedge
Delta exposure sums the delta of every option position, weighted by open interest and 100 shares per contract. It tells you how much an options book gains or loses when the stock moves $1. A dealer who hedges fully holds stock in the opposite direction, so the book's delta and the hedge cancel out.
The table below is the whole idea. It shows the option's delta from the dealer's side, and the stock hedge that offsets it.
Dealer long a call: positive delta, hedged by being short stock. Dealer short a call: negative delta, hedged by being long stock. Dealer long a put: negative delta, hedged by being long stock. Dealer short a put: positive delta, hedged by being short stock.
Holding a hedge versus trading one
A dealer may already hold the hedge it needs. New flow happens only when the book's delta changes: from new trades, a move in price, a change in volatility, or time passing.
Example: a dealer is short 1,000 puts with a delta of −0.30 each. The book's delta is +30,000 shares (short a negative-delta option is positive), so the dealer holds 30,000 shares short as a hedge. If the stock falls and the put delta moves to −0.40, the book's delta is +40,000 and the dealer sells another 10,000 shares. A large existing hedge is not continuous selling; only the change is a trade.
Who is on which side
Investors buy index puts for protection more often than they sell them, so dealers tend to be short index puts (Gârleanu, Pedersen & Poteshman, 2009). A dealer short puts has positive delta and holds a short hedge in the index, which it adds to as the index falls.
Caveat: that is a tendency in index options, measured over a past sample, not a rule for every stock or every day. Delta exposure is only as accurate as the assumption about who holds which contracts.