The electron on an atom gets excited to a higher level when some how the energy is transferred to the electron. But I can't understand it.
The way we currently understand in physics this interaction is exactly like that: a photon transfers its energy to the atom and as a consequence one of the electrons goes to a corresponding exited state. And this can only happen if the photon carries enough energy to make the electron jump at least the smallest gap in energy, otherwise they will not interact at all.
This is how things happen and we have no detailed information on the dynamics of this process, that is how it evolves in time. Theoretically, we have Schrodinger equation which could tell us how the process evolves in time, but we cannot check this experimentally, except in a few cases. For example some particles might oscillate between two states (see Neutral particle oscillation) and we have verified this change through time-scales large enough for our measurements, but has not been done for the case of the photon-atom interaction. We can estimate the time it takes for the interaction to occur, using Heisenberg's Uncertainty Principle but that is gives us no information on the time evolution of this process.
What happens when the electron in an atom is subjected to the electric field of energy equal to half that of the photon which can excite the electron to a higher level for a very short interval of time?
Well for this you need the mathematics and the knowledge of physics which I assume you don't have from what your question, but I can spare you the details and the specifics: We always solve the static case, the use these solutions to approximate the dynamic case to a solution expressed using them. All of this can be done when the time scales are very large and can be verified experimentally, but although in principle we could do it for very small time scales, at least theoretically, we cannot verify this experimentally. This leaves us in a state where we trust the predictions of Quantum Mechanics in this cases where no experimental validation can be physically achieved (see section on Experiments).
What happens when the electron in an atom is subjected to the electric field of energy equal to half that of the photon which can excite the electron to a higher level for a very short interval of time? What happens for the same conditions with magnetic field replacing electric field?
So when you put either an Electric or a Magnetic field, you get a series of phenomena like bending or twisting the atom, or even reshaping of the electron cloud (see the experiments section mentioned above). But remember, for smaller scales, we cannot say experimentally.
These questions are interesting, but unfortunately still out of our reach, at least for now. In my opinion, some research is needed to find a correct interpretation of Quantum Mechanics (see Wikipedia) because the fact that we have many of them, all predicting or describing the same experimental results, shows that part of the knowledge which differentiates them is missing and is the key to further understanding. Studying them and providing experimental tests is the only way to clarify questions like the ones you ask.