How they did it? (3D-like screenshots with animation) Here sample: 
                
             (from 20th second)

It clearly 3D render, as here we can see DoF effect with small defocus.

Also, it clearly not 2D transformations of screenshot, as some elements must be pixelized in closeups. But fonts always clear, even when it becomes large.

I think, they re-created every screenshot as complete scene in some editor like 3dsmax, but it must be very time-consuming to re-create every single element, redraw everything as 3D-model, add every text label and so on. But "window" looks totally flat, they can easily improve quality if they can spend so many time on it.

May be here some dedicated software/plugins for such effects? Maybe some tools to convert 2D screenshots to complicated scene?
This looks like 2.5D to me, probably in After Effects. You create high res art, perhaps by capturing web pages from a 4K display, then animate each screen. Its only a small amount of mousing. Then use the After Effects 3D camera to spin objects and place them in a depth field. turn on Motion Blur and add shallow DOF with a defocus effect driven by depth.