There are a lot of different ways it might be possible to detect artifact signals from ETI, but they are all rather speculative, and it might be difficult to distinguish them from natural phenomena, even if we did detect them, or they might not be repeatable. If ETIs are deliberately trying to advertise their presence to others, the most logical method would be to modulate a signal which occurs naturally in such a way that it's artificial nature is obvious, and to exploit the fact that other civilizations, would probably be listening at that frequency anyway, for other reasons.

This is what has always been so appealing about microwaves. Other cultures, whether they have SETI ambitions or not, would be doing routine astronomical observations anyway so a transmitting beacon at an astronomically important wavelength would have a good chance of being picked up by someone. The same can be said for harmonics of astronomically important wavelengths, which would be the first place SETI-ambitious cultures should monitor. The fact that these frequencies are all in the microwave spectrum, and that the interstellar medium is very quiet and transparent at those frequencies, makes them ideal places to listen, regardless of the technological level of the participant civilizations.

How do we know other cultures have astronomical technologies of this sort? We don't, but only the ones that do have any chance of doing SETI anyway. Highly advanced cultures without technology or with litttle interest in astronomy may be very common, but unless we drop in for a visit, we'll never know anything about them.

As to WHY we do SETI, I can't speak for anyone but myself, but I know why it fascinates me, and that's probably why it fascinates others.
A verified ET discovery would confirm that life and consciousness are not an accident or a miracle, but a logical development of, and a repeatable phenomenon in, a purely materialistic Universe. I.e., life is not a cosmic accident, it arises directly from the structure of the cosmos and is a consequence of it. If we are not alone, that means we are not special, either. If this sounds like a religious argument, you are absolutely right. It is a means of confirming a suspicion we have about the universe, and unlike traditional religions, it is one which can be proven.

Can it be disproven? Unfortunately not, if we never detect ETI, that is not proof of their nonexistence. The universe is very big and mostly empty. The only conclusion we can draw is that they are not common. My own personal opinion (I have only my own intuition to back this up) is that there is ETI, that it is extremely rare (perhaps only 10 or a 100 civilizations in the Galaxy at any one time, and that we will probably never find them. But we'd be crazy not to look, especially if we can piggy-back our SETI onto other scientific programs so as to justify their modest expense.