It makes sense that he should be more present in the supernal realms, given his ascent there, but in what sense is he more present in the physical world? The answer is…that the life of a tzaddik is not corporeal life, as is known, but spiritual life, which is faith, and the fear and love of G-d. * Source materials compiled by Rabbi Shloma Majeski. Translations are in bold. Underlining is the emphasis of the compiler.
Arranged and translated by Boruch Merkur
Having established a clear source in nigla for the eternal life of the tzaddik – for even in death he is said to be alive – it remains to be discussed in what manner that is so. Rabbi Majeski in Likkutei Mekoros cites from the inner dimension of the Torah, selections from Tanya describing how the tzaddik lives on even in the physical world:
It states in the Zohar HaKadosh that “The righteous who pass on are present in all the worlds more than when they were alive.” That is, even in this world of action [the physical world, regarding which it is said], “today, to fulfill them [i.e., the Mitzvos] (and tomorrow to receive their reward)” – the tzaddik is more present, for the actions [he did and inspired others to do and continue doing] grow, and give rise to ongoing generations of “offspring,” from the “light that was planted for the righteous” in the “field that was blessed by G-d [i.e., Gan Eden].” This light shines upon the land and the outer limits, reaching even “us [we] who are here today; we are all alive,” living in his ways, “the holy way it shall be called.”
The tzaddik’s presence is increased with regard to service of G-d, Heavenly matters, as well as to worldly matters. Indeed, it explicitly says in the Zohar HaKadosh that the righteous protect the world [the physical world], and their protection is more prominent posthumously than when they were alive. In fact, were it not for the prayers of the righteous in Gan Eden, the world would not continue to exist for a single moment. And all those who were close to the “sanctuary of G-d” in his lifetime are first to be blessed.
An elucidation of the above:
It states in the Zohar HaKadosh that “The righteous who pass on are present in all the worlds more than when they were alive, etc.” But it remains to be understood: It makes sense that he should be more present in the supernal realms, given his ascent there [upon his histalkus], but in what sense is he more present in the physical world? The answer is in accordance with what I was taught on the words of our Sages, of blessed memory, that “He dispenses life to all the living”: the life of a tzaddik is not corporeal life, as is known, but spiritual life, which is faith, and [the] fear and love [of G-d].
(Tanya, Igeres HaKodesh 146)
THE G-DLY SOUL: 
HIS ENTIRE IDENTITY
The following segment provides insight into what is different about the tzaddik’s eternal life given that all souls are eternal. The tzaddik lives eternally in a way that surpasses others on account of his G-dly Soul – his true life being spiritual life, as above – for that is his entire identity. Indeed, he has totally transformed his “Vitalizing Soul…having been transformed into good and literally subsumed within the sanctity of the G-dly Soul.”
It says about the G-dly Soul, “The soul that you have placed within me is pure.” It specifically says, “placed within me,” indicating that the person himself is not the G-dly Soul. The exception to this rule, however, is tzaddikim, the righteous. For with regard to tzaddikim, it is the reverse: the pure soul, which is the G-dly Soul, is the person himself, whereas his body is called “the flesh of man.” This reversal is reflected in the saying of Hillel HaZakein to his disciples: when he was going to eat he would say that he is going to do an act of benevolence to the despicable and impoverished one, referring to his body, for it was considered as something foreign to him. In this sense he does “an act of benevolence” to it by feeding it, insofar as he himself is only the G-dly Soul and nothing else. The G-dly Soul alone enlivens his body and flesh, because in tzaddikim, the evil that had existed in the Vitalizing Soul, invested within the blood and flesh, has been transformed into good and literally subsumed within the sanctity of the G-dly Soul. However, with regard to a Beinoni, since the essence and inner quality of the Vitalizing Animal Soul, which stems from the realm of evil, invested within his blood and his flesh, has not yet been transformed to good, the Animal Soul is the person himself.
(Tanya, Likkutei Amarim 37a)
Perhaps it is on account of the fact that his eternal “G-dly Soul alone enlivens his body and flesh” that the presence of the tzaddik in the physical world endures after death.