The Joseph Lee Comet is not native to this universe. It was brought into our solar system through an inter-dimensional rift centuries ago. This rift was caused by an astronomical explosion in another universe - a realm where physical laws as we know them do not pertain.
The comet emits a radiation that is undetectable to human perceptions and technology. Since the comet is from another universe, this radiation is not governed by the same laws of physics that radiation native to this universe is. When the Joseph Lee Comet passes the Earth it showers the planet with the unique radiation, which has no harmful effect on Earth's life and in less than a day dissipates into the vacuum of space. However, there are a handful of people who are permanently effected by the radiation. Due to a genetic anomaly these people possess a unique protein which the radiation binds to. However, this can only happen at a specific time. The inside of the uterus prevents the binding from happening and the cumulative effects of the Earth's biosphere on a person's body after only a few hours will prevent it from ever happening. All of this means that those few people who can be effected by radiation from the Joseph Lee Comet will only be effected right after birth and the exact day the comet's radiation is showering the planet. In other words, they have to be born during the day the comet is passing by the Earth.
What the imbued radiation does is interact with the quantum structures of the complex neurons of the human brain to cause the functioning nervous system to produce a very special energetic field that exist as an energy-based lifeform. The lifeform permeates a person's entire body and takes form as an inconspicuous energy pattern linked to the natural electromagnetic field of that person's body. Since this energy lifeform is created by a radiation of non-Einsteinian physics, it has two aspects that defy the conventional laws of physics. The first aspect is that it is tuned into and casually perceives the universal field that binds the superstring structure of the universe together, which gives it something of a universal omniscient awareness. The other aspect is that the energy lifeform can actually manipulate the structure of the universe, specifically the space-time continuum; although only in relation to the person the lifeform inhabits. The specific manipulation the lifeform is capable of is a warping and folding of time itself, which actually induces time travel. Because of the lifeform's connection to the universal field, the time travel actually involves not only a warping of the space-time continuum but also a fundamental alteration of reality itself on a universal scope. This altering of reality means the laws of physics and even causality simply don't apply in the time travel process. Therefore, this time travel allows the changing of history and can even result in paradoxes.
The time travel occurs only for certain reasons. This is because of the nature of the lifeform's consciousness and its universal awareness. The sentience of the lifeform is tied into its host's own consciousness and because of that, its actions are based on the person's superego and innate human empathy; the lifeform has a social consciousness. As a result, at some point in the person's life, the lifeform will start to regularly transport them into the past in order to help change the outcome of people's lives for the better. Each series of 'jumps' into the past follows the life of a person whose destiny the time traveler is meant to change. The first jump is typically several decades into the past; through a series of jumps, the traveler jumps nearer to the present, usually years at a time. The lifeform knows about these events and the ripple effects of changing them because of its universal awareness. It also feeds the objectives of the 'missions' to the time traveler through subconscious programming, which manifest in the form of a 'gut feeling' of what they are supposed to accomplish.
These 'shifts' through time occur seemingly at random. The time traveler is missing from the present for a period of time unrelated to the period he or she spends in the past. The only indication a person has to an oncoming jump is a feeling in their head which varies from right before they jump to several seconds of warning. They do not arrive in the same location they departed, but instead arrive near the person they are supposed to help. The jumps are typically restricted to the region they left from, usually keeping the time traveler in the same city and its surrounding area.