Both of those examples don't charge anything, and don't have ads on their wedsite/app. Both of them have (to the best of my knowledge) gotten hugh amounts of money in funding rounds. How exactly are they making money and why are VC's willing to invest?
It's not necessary that they are currently profitable to have received investment.
The reasoning behind the funding goes that, if a service is popular and has gained traction, a business model can be implemented later. This is particularly common amongst internet services with a strong network effect.
For this type of business, the purpose of the investment is primarily to allow it to scale, and survive through the early days while there is little or no revenue coming in. The goal is to 'get big fast', and gain widespread adoption. This is particularly important for network-effect services, as the value of the service to each individual user scales with the number of users on the service as a whole.
If a competitor came along and grew faster, it would encourage your users to switch (as the competitive service, being more pervasive, would be more valuable), and this loss of users would further drive down your relative value, and so on. Result: when you add value by being bigger, you need to try and be the biggest. Symmetrically, by growing big fast, you add significant barriers for any potential competitors.
Joel talks about this in his first strategy letter, with specific reference to Amazon.
There's more to explain though, because Amazon has a pretty clear revenue stream. Why grow big and own the whole market of a free service? The reason is that popular services are thought to be easy to monetise.
Revenue streams for these types of businesses include:
With specific reference to your examples:
Twitpic does have advertising on its pic display/comment pages. If you have adblock, you may fail to see many a business model.
There's an article here suggesting 10 ways to make instagram profitable.
In response to "why are VC's willing to invest?", read "The New New Thing" by Michael Lewis.. Roughly, during various bubbles, Silicon Valley funding is easier to get without a revenue history. Having a history, even a successful history, limits the imagination.