I've thought about this a lot, here's my best answer as of today.

Part 1 - Existence proofs

We think we can build 'intelligent' machines because human intelligence exists. We don't have a perfect idea of what it means to be intelligent, but we know, whatever we have exists so it's not impossible to create. Some disagree with this stance on philosophical or religious grounds, and others resort to pointing out even if it's possible it's really really hard. Fair enough.

Part 2 - Methods

Usually we start by picking the thing we think best typifies intelligence and we work to solve that. Chess, Written Language Comprehension, Visual Pattern Recognition, Spoken Language, etc. These are all called AI. All well and good, but so far we get exceptional single purpose systems (the narrower the domain the better) and then look around and say 'but that's not really intelligent.'

Lately people have started to think more about the process by which things become intelligent rather than just the end behavior. Somehow the machine should naturally transition to being intelligent as opposed to being explicitly programed with intelligence. Machine learning is so-hot-right-now because of this. But, again, in the end you get a well trained system that does whatever it used as it's error metric. Like differentiate cats and sailboats. "But that's not really intelligent because it can't [play mozart/read poetry/paint a picture]."

There are infinite criticisms, but they can be summed to a lack of 'generalness.'

Part 3 - Representation

What people searching for 'intelligence' are looking for is a system that can process data from at least as many sources in as least as many contexts as a human. The hard part there, and the one thing the brain does really really well, is being able to relate sight to sound, touch to taste, past to present, and present to future. In us there is a shared language of representation that encodes experience.

In AI so far it's an unusual system that tries to relate many senses, keep a life long memory, and work in a noisy and incomplete environment and constantly make predictions about what will happen next.

Part 4 - Data

It is an unusual AI researcher who has all the data they want. As computer people we are impatient, and so waiting 30 years for a robot to collect the 1.4 PB of visual information a human does by that age, or the 1.8 TB of audio information just isn't done. We use existing datasets that are computationally tractable (meaning you can run them in minutes, hours, or days).

And yet we do not have an existence proof that intelligence of the general human-like kind can exist without years of exposure to the world. It's reasonable to expect that it's possible, we just don't have pre-existing knowledge of that fact.

Part 5 - The Future

So how will we get there from here? We're probably going to have to do it the hard way. Create something that can sense the world in the ways that we can comprehend, and painstakingly rear it, collecting data, automating and hard-coding what we can, until we have a set of error metrics, motivations, data, and environment in which we begin to see the thousand different skills called 'intelligence' that humans take for granted. In short, we'll probably end up thinking a bit more like parents, and a bit less like computer scientists.