Assalaamu alaykum wa rahmatullaahi wa barakaatuhu. I read the following hadith:
Prophet Muhammad, sallallaahu ‘alayhi wa sallam, said, "Do not accompany except a believer, and do not serve your food except to a pious person." [Abu Daawood and At-Tirmithi]
I want to know about its authenticity. Andf it is authentic, then could you explain it, please? I am confused regarding accompanying the non-Muslims. Like, is it wrong to stay in their company or travel with them, and so on? And what if one gives food or serves food to a disbeliever? Is that wrong? Please help me to understand the hadith, and especially the two points I have asked about. May Allaah reward you, Shaykh.

All perfect praise be to Allah, The Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallahu ‘alayhi wa sallam, is His slave and Messenger.

The hadeeth which you mentioned was narrated by Abu Daawood, At-Tirmithi, Ahmad, and Ibn Hibbaan on the authority of Abu Sa‘eed, may Allah be pleased with him; they related that the Prophet, sallallahu ‘alayhi wa sallam, said, "Do not accompany except a believer, and do not serve your food except to a pious person."

The wisdom behind the prohibition of keeping the company of the disbeliever is that it yields love and affection, and Islam forbids the Muslim from having love and affection towards the disbeliever. Allah, The Exalted, says (what means): {You will not find a people who believe in Allah and the Last Day having affection for those who oppose Allah and His Messenger, even if they were their fathers or their sons or their brothers or their kindred. Those - He has decreed within their hearts faith and supported them with spirit from Him. And We will admit them to gardens beneath which rivers flow, wherein they abide eternally. Allah is pleased with them, and they are pleased with Him - those are the party of Allah. Unquestionably, the party of Allah - they are the successful.} [Quran 58:22]

Rather, keeping the company of people is usually fueled by love and affection and most likely results in being influenced by them. Allah, The Exalted, says (what means): {And do not incline toward those who do wrong, lest you be touched by Hellfire...} [Quran 11:113] Al-Qurtubi wrote:

"The verse indicates the command to abandon the disbelievers, sinners, committers of religious innovations, and the like because keeping their company is either Kufr (disbelief) or sin, since companionship results from affection. A wise man once said, 'Ask not about the person's character; rather, ask about the character of those whose company he keeps (and you can easily know what type of a person he is). Indeed, the person tends to be influenced by his companion and follow his example..." 

The prohibition of serving food to other than the pious applies to the invitation to a meal, meaning, do not invite the impious and feed him, because inviting him and serving him food may lead to developing familiarity and affection towards him. However, there is no religious impediment to offering food to a non-Muslim who needs it, like in the case that he is poor or a prisoner of war. The Companions used to serve food to the non-Muslim prisoners of war, and Allah commended such an act (in the Quran). 

'Awn Al-Ma‘bood reads:

"Al-Khattaabi underlined that this (hadeeth) applies to invitations, not to needed food; Allah, The Exalted, says (what means): {And they give food in spite of love for it to the needy, the orphan, and the captive.} [Quran 67:8] It is known that their prisoners of war were neither believers nor pious. The Prophet, sallallahu ‘alayhi wa sallam, is warning Muslims from keeping the company of other than the pious and forbade mixing with such people and eating with them because this often yields love and affection in the hearts..."

In brief, the meaning is: do not keep the company of other than the pious and devout Muslims. Do not take them as intimate friends whom you eat with and spend a lot of time with. You should know, dear brother, that every friendship in this worldly life will turn into enmity on the Day of Judgment, except for the friendship of those who are pious. Allah, The Exalted, says (what means): {Close friends, that Day, will be enemies to each other, except for the righteous.} [Quran 43:67] Ibn Katheer commented on this verse, saying, "The verse means that every friendship and companionship that exist for a purpose other than for the sake of Allah will turn to enmity on the Day of Resurrection. Only the friendship and companionship for the sake of Allah will last forever..."

For more benefit, please refer to fatwa 88293 about the criteria governing the relationship between Muslims and non-Muslims.

Allah knows best.