Divine influence is present in the world, and during the tzadik’s lifetime it affects his very body. Even after the tzadik’s passing, the divine influence is present at his grave and it is worthwhile praying at the grave and tapping into that divine influence.

Not only during the lifetime of the righteous, but even after their death, the divine influence stays with them and is present in some way at their graves. For their bones, which [during their lifetime,] were already vehicles for the divine influence to take hold, still retain within them sufficient distinction to bestow that kind of influence. This is the reason why our Sages, of blessed memory, said that it is worthwhile prostrating oneself at the graves of the righteous and praying there to God. Prayer there is more readily accepted [than in other places], given the presence of the [physical] bodies which had already been affected by divine influence. (Rabbeinu Nisim Gerondi, Derashot HaRan 8)

There are five levels of spiritual achievement when visiting graves of the righteous. The highest levels include the contrition and self-negation a person feels when encountering the soul of a righteous person, as well as connecting one’s own spirit with the spirit of the tzadik.

A person feels contrition when encountering a tzadik:

People may visit the grave of a tzadik that they knew and were familiar with during his lifetime…when he was within a physical body. But now that the tzadik’s soul is in its pure spiritual state, this can bring the visitor to experience great contrition and actual self-negation, for he feels extremely ashamed by all of his actions and thoughts until now – since they are all revealed. For even while he was alive the tzadik knew people’s thoughts and plans, as is well known. This is certainly the case for the soul of the tzadik after he passed away, which is purely spiritual.

Clinging to the soul of the tzadik:

Furthermore, through a powerfully connecting belief in the [greatness of the] soul of the tzadik [which comes] from the innermost part of a person’s heart, and through using one’s spiritual powers of imagination and creativity, and through great self-negation, a person’s soul can cling to some aspect of the soul of the tzadik who dwells there [in the grave]…. This is the concept of belief in the righteous, whereby people can travel to visit their graves even though they do not grasp the tzadik’s spiritual level at all. (Rabbi Shalom Dovber Schneersohn, Kuntres HaHishtatĥut 8)

Rabbi Yitzĥak of Neskhiz had opponents who would annoy him in various ways. Once, when they intensified their pursuit of the rabbi, one of those close to the tzadik was unable to hold back and cursed them.

In every generation sparks descend from the soul of our teacher Moses, may he rest in peace, and they clothe themselves in the bodies and souls of the sages of that generation, the “eyes” of the community, to teach the people so that they may know the greatness of God and serve Him with their heart and soul. (Rabbi Shneur Zalman of Liadi, Tanya, Likutei Amarim, chap. 42)

Through his holy actions a tzadik can cancel all [negative heavenly] decrees against Israel. When the tzadik issues a decree, the Holy One, blessed be He, fulfills it. As the Sages state: The tzadik decrees and the Holy One, blessed be He, fulfills (see Ta’anit 23a). (Rabbi Elimelekh of Lizhensk, Noam Elimelekh)

The tzadik always stands between God and the world, connecting the mute and dark world with divine speech and divine light. (Rabbi Abraham Isaac Hakohen Kook, Orot HaKodesh 3:229)

The pure tzadikim do not complain about evil – they instead add righteousness. They do not complain about heresy – they instead add belief. They do not complain about ignorance – they instead add wisdom. (Rabbi Abraham Isaac Hakohen Kook, Arpelei Tohar, 39)

Further reading: For our Sages’ comments about the great levels the righteous reach, see A Concise Guide to the Sages, pp. 44, 179, 211, 337.