remember that rape is still exceedingly common (1 in 3 women in the USA will be sexually abused at some point). women are just as sexual as guys, it's just much more dangerous (and socially unacceptable - slut-shaming, etc) for them to solicit sex. as for why anonymous dudes on the internet are so creepy, it's most likely a combination of the disinhibition anonymity gives you and the sexual deviance a boner gives you.
Once one person realizes it's a successful strategy to send more messages then their competition, their competition also has to send more messages. Arms race quickly ensues. And once you have such an arms race on one side, the other gender no longer has to even bother initiating conversations.
Why did the arms race tip towards men competing and women getting it easy? Probably cultural norms. But once the arms race got started, there was no putting it back in the bag.
Cultural norms that are followed by every live-birthing animal species, how strange.
(Put another way: your explanation would cover our closest relatives: chimps, gorillas, orangutans and bonobos – given the huge differences in behavior, a simple reproductive cost model is obviously incomplete)
This might initially seem like a silly question, but as someone who wants to understand the perspectives of both sexes I wonder: Why can this behaviour generally be observed only in male to female exchanges but not the reverse?