Ok, I'll do my best. Warning: the numbers in this are not accurate as I don't know all the true numbers, but they should give you an idea of what the issue is.
All tests can give you a false positive or a false negative, but with some it's more likely than others. Urban legend suggests the LFTs have a high-ish false negative rate. I'm going to use 10% for this- so that would mean that if you have Covid, you have a 90% probability of a positive LFT and a 10% probability of a negative one. (I suspect it's actually less than 10%, but 10% is a nice easy number to work with.)
Now, what you actually want to know is not the rate of false negatives, but what the chance is that you do in fact have Covid even though you've had a negative test result. And that depends on how likely you are to have Covid in the first place. So in the general population, a certain proportion have Covid today - I haven't looked this up but it's in the order of magnitude of 1%. So let's say, in the absence of any further information, there's a 1% chance you have Covid right now.
If you've had a positive PCR a few days ago, the chance you have Covid right now will be quite a lot higher. I really don't know what this is, but for ease of calculation let's say it's 90%.
Now, take 1000 people from the general population. Of those, we'd expect 1%, that is 10, to have Covid. Of those 10 people, 90% (or 9) will test positive and 1 will test negative. If, for the sake of making the maths easier, we assume that there are no false positives, all the 990 people who don't have Covid will also test negative. So we have 9 people who've tested positive, and 991 (made up of 990 people who don't have Covid, and one who does) who've tested negative. That means, if you have a negative test result, your chance of being the 1 false negative is only 1 in 991, or about 0.1%. You don't really need to stress too much about this, it's a pretty low chance.
Now, take 1000 people who've had positive PCR tests in the last few days. We're going to assume 90%, or 900, of them are in fact positive. We test everyone with an LFT, and 90% of our positive cases, or 810 of them, test positive, while the remaining 90 test negative. We're going to assume again that all the 100 people who are negative test negative. So we've got 190 people who've tested negative, of whom 90 do in fact have Covid. So now, if you get a negative result, your chance of having Covid is 90 out of 190, or about 47.4%. That's a much higher chance that you do have Covid, even though your test says you don't.
I hope that makes sense. It's easier to explain to people in person when I can draw diagrams!