Writing is another common term for "selling." They mean the same thing. Writing a put is selling a put. The best way to visualize this is that for every buyer of a put option, there has to be another party that sells that put.
Let's walk through a transaction, but we'll start with a covered call because it is easier to understand: In a covered call trade, the trader buys shares, believing they will go up (duh) and then also sells a call against them. That is, she buys IBM at $130 a share and then for $4 a share, sells call options with a $150 strike which expire in 4 months. If IBM rockets upwards to $180 in the time period, the trader makes $20 a share from the shares that went from $130 to $150 - and also makes $4 from the call option she sold for a grand total of $24 in gains. But if she had just owned the stock and played golf the whole time, she would have made $50 in just owning the stock, nothing fancy.
In a covered put, things happen in reverse. She shorts IBM at $130, thinking it's going down like a horny teenager on a Saturday night. But she is nervous so she sells put options to go along with this trade. Here, she might sell puts with a strike price of $110 strike for $4, which expire in 4 months. That is, the buyer of the put has the right to "put," or sell, IBM back to our friendly female trader here at $110 a share. So if IBM tanks amid fraudulent accounting rumors and goes to $80 a share, our trader makes $20 a share from the decline from $130 to $110. And then she pockets another $4 from the put premium she sold. At $80, her IBM shares will be put to her but she will be "covered" because she has the shares that she shorted in the first place. That is, the risk is less. And the reward is less too: If she'd just shorted IBM and played golf, she would have made $50 from IBM's decline from $130 to $80 in that time period.