Body Count was an aspect of the attrition based war that took place in Vietnam.
US Military intervention initially prioritized propping up the unstable South to create a strong buffer against reunification efforts from the North. To invade the North risked involving stronger Communist powers into play. This started with deployment of American "advisors" who essentially trained ARVN (Army of the Republic of Viet Nam) soldiers, accompanied them on missions and played a support role in the South. True combat soldiers were eventually deployed in a defensive mission to protect air force bases from Viet Minh raids.
The problem became apparent that ARVN soldiers were not effectively repelling North forces. After suffering heavy casualties in numerous engagements with the North, desertion rates increased and overall morale plummeted. It's important to note that ARVN soldiers were stereotyped for being ineffective and cowardly; already a poor reputation. Their suffering track record and string of heavy losses exacerbated the sense that the South was in fact losing the war.
General Westmoreland of the US rallied to turn the tide and stem the losses suffered in the South by making US involvement more open ended. He reasoned that US military strength existed primarily in its offensive capabilities because of the nature of US military schooling. His plan was to "seize the initiative" to destroy guerrilla and organized forces of the North (read: search & destroy). This phase of engagement would require the continued commitment of the US (keep giving us money, soldiers and weapons) and would end when the enemy was thrown on the defensive, exhausted, low on resources and unable to launch insurgent or organized attacks. (Read: go on the offense by seeking out enemy forces within South Vietnam, but don't actually invade North Vietnam)
Essentially Westmoreland reasoned that US superiority in warfare technology and resources would eliminate more bodies than the North could replace. Part of his attrition strategy did also include recruitment strengthening in the South (primarily to discount another body for the North). American soldiers and technology would advance to wipe out the bulk of the North Vietnamese Army's (NVA) organized forces and ARVN would be relegated to taking over defensive roles in the South and bolstering recruitment. Attrition was to be supported with heavy firepower and bombing that would in theory suppress enemy safeties and hiding spots.
Here; after all that explaining is why Body Count was the favored metric of success. The body count concept affirms Westmoreland's strategy of attrition and gave tangible "evidence" of winning the war. The lack of fortified positions to take over in the brush of Vietnam and the extremely rugged nature of the terrain made it difficult to really say: "this dense jungle is controlled by us" or "this air force base is now controlled by us". However, Body Counts almost always sounded favorable to the US and gave the illusion that the US was winning.
Ultimately Body Counts proved to be problematic because of easy manipulation. Leadership would sometimes instruct forces to add innocents and civilians to the days body count. Pure inflating of the numbers reported also occurred. Morevoer, the North was able to control the rate at which they suffered losses by avoiding large scale confrontation by organized forces (which is what the US army excelled at).
I wont speculate as to whether or not attrition could have won the war for the US or even if it definitively lost it for them. That is an intensely debated topic today with many schools of thought arguing for or against.
However the tl;dr of your question about Body Counts is that Body Counts were used as (biased) evidence of the success of attrition. Attrition was chosen as the strategy in Vietnam because of the end goal of creating a strong buffer against the North, rather than all-out invasion.
I found George C. Herring's book America's Longest War: The United States and Vietnam, 1950-1975 to be a good comprehensive history of the conflict with good detailing of American side decision making.
US involvement in Vietnam was ostensibly to protect the South Vietnamese state. to engage in an offensive war against North Vietnam was seen as politically risky since the US did not want to drag China or the Soviet Union into the conflict. as a result the US strategy was largely limited to a defensive and/or retaliatory role against North Vietnamese invaders or Viet Cong guerrillas.
the combination of geopolitics and geography meant that there were no clear "front lines" in the conflict, as the US military was familiar with from its experiences in WWII and Korea. traditional measures of military progress such as capturing territory or taking a hill were practically meaningless. without this traditional means of gauging success, the US Army high command had to find a new way of reporting their strategic situation.
the "body count" strategy was implemented in the aftermath of the Battle of the Ia Drang Valley in November 1965. the battle featured a new experimental US doctrine where ground troops were inserted via helicopter into areas with heavy enemy presence and by virtue of airborne support and overwhelming firepower were able to inflict tremendous casualties against superior numbers with a relatively small force. at LZ X-Ray (the first of 3 separate combats in the Ia Drang battle) the Americans suffered 79 killed and 121 wounded, in comparison to over 600 confirmed and possibly over 1000 NVA killed in action. the "success" of the Ia Drang battle convinced Army commanders that small numbers of US infantry supported by air power and artillery could inflict massive casualties on the enemy far in excess of their own.
US Army commanders seized on this lopsided "victory" as a convincing method of reporting progress in an otherwise chaotic and unmanageable conflict. it looked good on paper, and gave politicians a means of presenting a more positive narrative of the course of the war to the American public. that it turned out to be a pitifully inadequate measure of the actual strategic situation and was rather limited in effectiveness was largely ignored until the later stages of the war.
Sheehan, Neil, A Bright Shining Lie: John Paul Vann and America in Vietnam
/u/Jwang909 has it summed up well -- it was an effective measurement tool that reinforced the strategy, gave an illusion of winning, and was frankly the best tool they had. I might add that all the services had similar metrics. The Air Force and Navy were somewhat obsessed with the numbers of trucks destroyed, bridges downed, etc., not realizing that 1) many stats were unverifiable, tended to inflate due to inter-service rivalries, and 2) Maybe those aren't the best stats for measuring effectiveness.
There's a wonderful book on this exact subject called "No Sure Victory: Measuring US Army Effectiveness" by Gregory Daddis. It answer your question in wonderful detail.