"And blessed are all they who do hunger and thirst after righteousness, for they shall be filled with the Holy Ghost."
This is not about hunger or poverty. This is about fasting and seeking after righteousness.
You qualify for this blessing by hungering "after righteousness." You qualify by thirsting "after righteousness." In other words, you receive the Holy Ghost in proportion to the hunger and thirst you display to receiving it.
Fasting is a promised means for increasing the Holy Ghost in your life. We read this about Alma when he served as High Priest over the church: "And this is not all. Do ye not suppose that I know of these things myself? Behold, I testify unto you that I do know that these things whereof I have spoken are true. And how do ye suppose that I know of their surety? Behold, I say unto you they are made known unto me by the Holy Spirit of God. Behold, I have fasted and prayed many days that I might know these things of myself. And now I do know of myself that they are true; for the Lord God hath made them manifest unto me by his Holy Spirit; and this is the spirit of revelation which is in me. And moreover, I say unto you that it has thus been revealed unto me, that the words which have been spoken by our fathers are true, even so according to the spirit of prophecy which is in me, which is also by the manifestation of the Spirit of God." (Alma 5: 45-47.)
Fasting and praying opens the Spirit. It allows you to know a matter through the power of the Holy Ghost. Again, we read this about the Sons of Mosiah who were completing their service as missionaries: "...Alma did rejoice exceedingly to see his brethren; and what added more to his joy, they were still his brethren in the Lord; yea, and they had waxed strong in the knowledge of the truth; for they were men of a sound understanding and they had searched the scriptures diligently, that they might know the word of God. But this is not all; they had given themselves to much prayer, and fasting; therefore they had the spirit of prophecy, and the spirit of revelation, and when they taught, they taught with power and authority of God." (Alma 17: 2-3) They not only searched the scriptures, but they also spent time praying and fasting, that they might show God their earnest commitment to know the truth. The result was the "spirit of prophecy, and the spirit of revelation." Or, in other words, they were filled with the Holy Ghost.
The Lord speaks in simple formulas. They work; when tried in sincerity, acting no hypocrisy, with real intent, they work. Half-hearted efforts are not so effective. But when a soul, any soul, hungers and thirsts after righteousness, they are filled with the Holy Ghost.
This sometimes presents a problem for those who have medical conditions which prevent them from fasting. In The Second Comforter, I've suggested there are other ways to subordinate the desires of the flesh as a way to "hunger and thirst" while keeping medical needs satisfied. Reducing calories, or doing without some other thing as a form of "fasting" can be substituted. The decision would be between you and the Lord, but there are always ways provided for meeting what the Lord asks, including fasting by those who are medically unable. (1 Nephi 3: 7.)
This sermon is a blue-print of the Lord's new charter for mankind. It is the new, higher way of living. It is intended to result in a new spiritual life for those willing to live it. Therefore you should not dismiss "hunger and thirst for righteousness" as something trivial. If you are among those who does not believe the Lord speaks with them, take these invitations from the Lord seriously. They are designed to reconnect you with God. They have the power to accomplish it.
Also, in the case of the Sons of Mosiah, there was actual "power" which came through this means. These missionaries could teach "with power and authority of God" because of their fasting, prayer and study of scripture.