Why do we say that trade and specialization according to comparative advantage lead to one-time increase in PPF of given country? PPF shows us different combinations of goods that we would be able to produce if we employed existing resources of our country in the most efficient way. 

One thing to notice about PPF: It's completely independent from our agregate utility function. For an example, if we have only two types of goods, wine and cheese, then maximum production of only cheese (like 100;0) will be on our PPF, even if we would want to have some wine too or if we don't need so much cheese. 

Now suppose we (Cheesetopia) trade with another country (Winetopia) and specialize in our comparative advantage (cheese). We will be able to consume, for an example, 50 cheese and 50 wine at the same time, a feat impossible for us under autarky. We would be consuming outside our production capabilities, outside of our PPF. But our current production would be on our good old PPF. 

Thus I fail to see how free trades causes any shift in PPF or causes a country to produce outside its PPF. At the least for the Ricardian model.
For this you first have to put some numbers on the problem to be able to draw PPF and consumption. Lets assume that Cheestopia can produce 1 cheese at 1 labor, and 1 wine at 2 labor and Winetopia 1 cheese at 2 labor and 1 wine at 1 labor, and both country has 100L avaiable for production, this gives you the following PPFs for both countries in autarky (the chart was created by 1muflon1):

 

Now, I wont post full solution for a Ricardian model because it is not part of your question but in this case valid solution would be that the trade prices will be 1 cheese for 1 wine (if you want to know why maybe post it as a separate question about solving Ricardian model). Now under this trade scenario both cheestopia and Winetopia can produce only the good in which they have comparative advantage (i.e. only cheese in Cheesetopia and only wine in Winetopia) and at the trade 1ch for 1 w they can now access all the following points in consumption (note the points for two countries overlap but I ploted them next to each other so you see that second country does not disapear):



Now, whether anyone agrees that this increases a PPF it is indisputable that it increases the consumption possibilities of two countries. Thats plain from the picture. So now a question is following. Can we call this expansion of consumption possibilities increase in PPF? I think yes and also many textbooks call it like that for the following reasons:


From economic point of view there is no difference between producing, say, X of wine and between producing Y of cheese and then trading Y of cheese for X of wine. Just imagine that there a wizard/machine that converts Y of cheese in X of wine. From an economic perspective trade is like having access to production technology of the other country. Also macroeconomically consumption must be in long run equal to income and income in the long run must be equal to production.
By standard measures of production (such as GDP) this increases the production at any other point than the vertex (100ch, 0w) which overlaps with autarky PPF and PPF is by definition all possible combination of production. To show that this is the case lets consider one point on the new consumption line and lets use one standard output/production measure - GDP.


Lets select Cheesetopia and lets select point of consumption (0c,100w) which is a point of consumption only accessible by trade. Now if you claim that this point does not represent new PPF, at this point the production then a consumption point (0c,100w) should not show up on aggregates as GDP which measures the production as that would imply that production is outside PPF which by definition must include all production possibilities of a country. To calculate GDP we have to choose some good as numeraire (money) it does not matter which one we choose lets do cheese. So at the consumption point 0 cheese and 100 wine the components of GDP will be C= 100ch (here ch is the currency cheese they actually consume wine), X = 100ch (they export cheese) and M = 100ch (they import 100 wine and again cheese is the currency here). So GDP in this case is GDP = 100 + 100-100 = 100. So here we have a contradiction and there is only two ways how to resolve this contradiction:


You will say that GDP does not measure production. Which I dont think even the biggest critics of GDP would say.
PPF must be at consumption point (0ch,100w) at least 100 because by definition PPF is the combination of all production possibilities of the country, so if GDP measures production it must be within PPF it cannot be outside.


Now you can pick your 'poison' from the above option. Economically, the second makes sense. Intuition is that there is no meaningful difference between producing cheese at home and via trade transform this production of cheese into production wine and between actually being able to produce wine at home at the terms of trade. Or another way how you could express that intuitively is that in the new PPF any combination of goods can be produced by producing just cheese which is the highest point on the pre-trade PPF and which is also the single point where pre and post trade PPF will be exactly same.

Furthermore, to see that this is the same lets consider this scenario. 1 Winetopia gets destroyed by a tsunami so only Cheesetopia is left in the world. Now imagine that Cheesetopia will send some scubadivers to recover the Winetopia's wine production technology hence now Cheesetopia can produce both wine and cheese at 1 labor. Now the PPF is exactly the same as with the trade (meaning there is no meaningful economic distinction between trading.)