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Hi guys, I found few circuits similar to that one (link below) but I can't find any proper explanation of working of such circuit other than very very general one. I'd like to know what exactly every element does. Could anyone help me with analysing this circuit ?

https://www.electronicsforu.com/electronics-projects/mobile-phone-detector-using-lm358

A traditionnal analysis DC then AC is still usefull. See the short description for the circuit supplied in the link.

What is not included is the role of each resistor. It should be evident from the DC analysis part, given that the LM357 (why using a 358 and letting half of it un-used? maybe bacause you don't have a 357 at had, otherwise, ... I just don't know) is not a rail to rail OpAmp and its input would be thus its output would be limited, oblidged, to be between 1 volt and 3.5 volt (from memory). The capacitors, at that stage, can be of any value since we are on a DC analysis and since no circuit loop need them to be completed, the resistors prevail.

For the AC analysis, that where things get intereting. First, we replace the capacitors by a full wire (and the DC source by a ground). Doing so, both inputs of the OpAmp become at the same voltage (idealy). Somehow, the initial drawing could have been a little bit better to make it obvious, but anyhow, if you don't immediatly see it, do it manually and you should get it. When that occur, for an ideal OpAmp, its output is a zero volt, which turns off the BJT. Momentarely. Indeed, the capacitors are not a full wire and they take time to charge and discharge.In particular, C2, which is the link between the two inputs of the OpAmp, will not charge as fast as C1 given the C2 is, in part, in parallel with R1. So, for short period of time, the voltages at the two inputs of the OpAmp will oscillate. Without caps, it would be too fast for the human eye perception and too noisy, taking any noice picked by the antenna.The caps C1 and C2 make the variations filtered and the OpAmp will then hopefully provide a blinking perceptible by the human eye. Anyhow, the OpAmp, without these caps, could probably not handle the frequency from cell phone waves (but I am NOT an expert in that domain).

Note that if you realize the original circuit and if it does not work "fine", try pushing the LED at a more traditionnal place, that is, given that you are using a NPN (or an NMOS), push the LED to the collector (drain) side instead of having it at the emitter (source) side.