The inside view refers to what your explicit models tell you. For example, if you write down a plan with explicit contingencies for things going wrong and time estimates for each step, or write down a careful argument for your point of view, or solve some equations. The outside view refers to what "an independent observer" might think; the proverbial man-on-the-street; a perspective which doesn't pay attention to all the details of the situation, but places it as an instance of a class. For a plan, you have some outside-view probability that it will fail or take longer than you expect, based on unforeseen complications rather than explicitly-listed contingencies. For an argument, although the argument may conclude a belief with some high probability, your outside-view probability should usually be lower to account for your uncertainty that the argument is correct. For the equations, the outside view accounts for the probability that you've made a mathematical error.
One way of thinking about inside view vs outside view is that they are human-friendly ways of thinking about the likelihood function and the prior. Somehow, humans are frequently tempted to commit base-rate neglect: the belief in a hypothesis is just the extent to which it matches the data. For example, if a test for a rare disease comes up positive. People are not inclined by default to multiply the evidence for something by the prior odds against, except in everyday circumstances which they've experienced a lot. (I think this mistake may also account for the conjunction fallacy.) "Outside view" helps you imagine base rates by envisioning a perspective with more distance from the issue. So, you come up with an inside view and an outside view and "multiply them together" to become a proper Bayesian. This gets rid of base-rate neglect, and related biases such as the availability heuristic (where you implicitly use "number of times I've heard about this" or similar things as a substitute for base-rates).
However, I think this underestimates the usefulness of outside view. Robin Hanson discusses it almost as if it is equivalent to eliminating bias. While I do think a strong outside view needs to be combined with a strong inside view, I think there's a sense in which outside view is the first thing you reach for to debias almost anything, not just base-rate neglect.
Say you learn about a new bias, such as omission bias, and you don't have any special debiasing tools for it. What you do is you file it away under your list of important biases. Then, when you encounter a situation which pattern-matches to the bias, you think something like "Ahh, but omission bias!" and attempt to adjust for it.
In other words, you have some kind of quick pattern-matcher which looks out for signs of the bias. When it fires, you may do a more thorough check: are you in the kind of situation where the bias occurs? If so, you attempt to correct for the bias. This is outside-view reasoning; you are noticing that the situation has the general flavor of one which gives rise to bias, and attempting to follow the general policy which would correct for that bias, rather than the biased policy which you would follow by default.
If all goes well, jumping to outside view helps you subsequently develop more targeted debiasing strategies which hit the bias directly, including an inside view which doesn't produce the bias in the first place. However, I want to give outside view credit for being the quick skill that you apply all over the place which helps to generate the further skills.