A "political opinion", at least in my parlance, is a system of beliefs which includes personal preference for a particular political party or ideology. "It is correct to adopt libertarianism" might express a political opinion. "The number of Jews killed in the Holocaust is about six million" is perhaps evidence that the speaker isn't a typical Nazi, but it is too narrow and concrete to be called a political opinion per se.
As for the reasons why the caution applies to political opinions while does not (that much) apply to simple questions of fact even if they have political implications, consider how many obvious fallacies one has to commit to give a wrong answer. "How many Jews were killed in the Holocaust" is a question of fact with pretty straightforward meaning. Every reasonable person would agree that one should look into the archives and population censuses. Practically the only way to credibly justify an incorrect answer is to postulate a conspiracy which has faked all official evidence - and this is what actually happens. Then everything you need is Occam's razor: conspiracies are complex and therefore very improbable.
On the other hand, when considering the question "is it correct to adopt liberalism", both "correct" and "liberalism" are open to wide range of interpretations and there is no universally agreed upon method of answering this sort of questions. The biases have much easier role here in their effort to sway the result towards the prejudiced answer.
There are some issues where a) I have no direct knowledge of the facts, b) there is some dispute over the facts, c) I hold an opinion on what the facts are. You probably do too.
For example, I believe that ~6mm Jews were killed in WWII, and that most people who deny this are anti-semites. If someone produced a theory which conflicted with my views, I would probably be very suspicious of the person's motives. I think this without ever having seen evidence that would satisfy me directly. [I've been to Auschwitz and a few other camps, and I've read a few books about WWII, but the only reason I have to cite the figure of 6mm rather than 3mm, or even 300,000, is that most knowledgeable people use that number]. I suspect most LWers are at a similar state of opinionated ignorance.
Now, I am in no way incentivised to investigate this: my opinions matter roughly zero to anyone, including me. So I don't have any reason to investigate the Holocaust. But there are other areas where the facts do matter to me.
I know very little about medicine. When I need medical assistance, I tend to do what a doctor tells me without criticising his diagnosis. For example, I suffered from eczema a few years ago. I was prescribed a medicine which contained hydrocortisone as the active ingredient. I Googled this ingredient and then took the medication until the affected area cleared up. I noted that there were other steroids available [Clobetasol propionate is one such] which are considered cures for eczema. I did not know why my doctor prescribed me one rather than the other, and the distinction between the two would likely affect me. But I took the 'expert advice' without a pinch of salt. Unlike the WWII question, here the facts are actually relevant to me, and a mis-diagnosis could have caused some complications. But just as I don't check the wiring of my house's electric systems, or check my walls' stability and capacity to support my ceiling, I take my doctor's advice.
There are some less clear-cut examples. For example, the 'smartest' [read: some combination of high-IQ, high-rationality, knowledgeable] people I know tend to identify as either libertarians or utilitarians, rather than as socialists. There are a few exceptions to this, but not many. From this, I could arguably assume that it would be correct to adopt a libertarian or utilitarian, rather than a socialist*, mindset to politics without actually understanding why libertarians are libertarian. Just as I don't need to know why my doctor prescribed hydrocortisone in order to take it, I don't need to know why most smart people I know favour relaxed drug laws in order to share their opinion.
*I'm aware that these are not the only options, but they're the most mainstream 'labels' which are clearly defined. ['Liberal' and 'conservative' mean different things in different countries, but I think those three are relatively constant, at least in the people who apply them to themselves]