The term “afflict your soul” or “afflict your souls” appears in several places in the Old Testament, in particular in the instructions for keeping the Day of Atonement.

Lev 16:29  “This shall be a statute forever for you: In the seventh month, on the tenth day of the month, you shall afflict [anah] your souls [nephesh], and do no work at all, whether a native of your own country or a stranger who dwells among you.

Lev 16:31  It is a sabbath of solemn rest for you, and you shall afflict [anah] your souls [nephesh]. It is a statute forever.

Lev 23:27  “Also the tenth day of this seventh month shall be the Day of Atonement. It shall be a holy convocation for you; you shall afflict [anah] your souls, and offer an offering made by fire to the LORD.

Lev 23:32  It shall be to you a sabbath of solemn rest, and you shall afflict [anah] your souls; on the ninth day of the month at evening, from evening to evening, you shall celebrate your sabbath.”

Num 29:7  ‘On the tenth day of this seventh month you shall have a holy convocation. You shall afflict [anah] your souls [nephesh]; you shall not do any work.

The Hebrew word translated “afflict” is “anah”, which means to afflict, humble, bow down.  The Hebrew word translated “souls” is “nephesh”, which means a living creature.

Are there any other Scriptures which help us understand what it means to “afflict your soul”?  The answer is yes.  There are several verses which associate fasting with affliction (anah).

Ezr 8:21  Then I proclaimed a fast [tsom] there at the river of Ahava, that we might humble [anah] ourselves before our God, to seek from Him the right way for us and our little ones and all our possessions.

Psa 35:13  But as for me, when they were sick, My clothing was sackcloth; I humbled [anah] myself with fasting [tsom]; And my prayer would return to my own heart.

Isa 58:3  ‘Why have we fasted [tsum],’ they say, ‘and You have not seen? Why have we afflicted [anah] our souls [nephesh], and You take no notice?’ “In fact, in the day of your fast [tsom] you find pleasure, And exploit all your laborers.
Isa 58:4  Indeed you fast [tsom] for strife and debate, And to strike with the fist of wickedness. You will not fast [tsom] as you do this day, To make your voice heard on high.
Isa 58:5  Is it a fast [tsom] that I have chosen, A day for a man to afflict [anah] his soul [nephesh]? Is it to bow down his head like a bulrush, And to spread out sackcloth and ashes? Would you call this a fast [tsom], And an acceptable day to the LORD?

The Hebrew word translated “fast” is “tsom”.  It comes from the Hebrew “tsum” meaning to “cover the mouth”.  So it means to not eat or drink anything.

There is also another verse which associates affliction of the soul to hunger.

Isa 58:10  If you extend your soul [nephesh] to the hungry And satisfy the afflicted [anah] soul [nephesh], Then your light shall dawn in the darkness, And your darkness shall be as the noonday.

Lastly there is one more verse about affliction of the soul, but it does not relate to fasting.

Num 30:13  Every vow and every binding oath to afflict [anah] her soul [nephesh], her husband may confirm it, or her husband may make it void.

Conclusion

Using Scripture to interpret Scripture we can see that to “afflict your soul” means to humble yourself and the preponderance of evidence is that this should be done by fasting, that is, abstaining from food and water.