That question is a bit difficult to answer. Let me try a few, hopefully one of them helps you.

1) It's there because it works. A large part of high-energy theory is what is called model building, where people build models that at the least agree with previous knowledge and are internally consistent. Hopefully they also explain something not explained before. Such a phi^4 potential term is very natural to try (cannot elaborate on that here), and gives the Higgs boson and mechanism as we know it, and thus the Standard Model.

2) The phi^4 is the reason for the self-interactions, agreeing with observation.

3) This one is complicated. The Higgs potential energy contains two terms, one proportional to phi^2 and one proportional to phi^4. In quantum field theory, terms quadratic in a field correspond to the mass term for that field. Without the phi^4, the Higgs would be a free, non-self interacting particle. Moreover, the whole reason why the Higgs field can give mass to other particles, is because its mass term *is negative*! This is called "spontaneous symmetry breaking", a hugely important concept associated with phase transitions. You can picture it as follows: a ball in a valley will roll to the bottom and stays there. The negative mass (potential) term however causes the middle of the valley to rise to a small hill (local maximum). The ball will now want to roll down away, but any direction is equally good, and one is chosen randomly or "spontaneously". The phi^4 term is necessary to make the valley curve upward again, so that the ball does not roll down indefinitely. Thus to have any model with spontaneous symmetry breaking (which agrees with the fact that the Higgs field permeates all of space), a phi^4 term is necessary to make the potential energy bounded from below.