Well, the resupply was not something that happened when Israel was close to defeat, at least not in the Sinai. It was ordered when the Israelis were facing a defeat, without a doubt. The airlift, which began on 14 October, had been ordered to begin on the 9th, and came after a decisive battle in the Sinai on the 14th, which may have made it superfluous in terms of "saving" the Israelis in the Sinai.

On 12 October, under increasing pressure from Egyptian General Ahmed Ismail Ali, Lieutenant General Sa'ad Al Shazly was given orders by General Ismail (over all objections) to make for the Gidi and Mitla passes. A "political decision" had been made, to relieve the pressure on Syria, and Shazly had his hands tied. He reminded General Ismail of what had happened last time a brigade was caught without air cover, as they would be (they were advancing past air defense ranges), but to no avail. An attack would be launched from the bridgehead. Shazly got it postponed 'til the 14th.

The IDF had been debating ending the war with a ceasefire, or undertaking a hazardous canal crossing, right around the 11th or 12th of October. It didn't think it would be able to continue this war, especially if it turned into a war of attrition. Then word came on the 12th that Egyptian armor was moving and crossing the Suez Canal, appearing to be preparing for an attack on the 13th or 14th. Israel decided to wait, instead of taking initiative or making peace, and to fortify positions.

Shazly put it this way: "The enemy had 900 tanks in his operational zone. We were attacking with 400. We were doing so, against well-prepared positions, in precisely the 'penny packets' that had cost the enemy so dear over October 8-9. And we were condemning our tank crews to attack over open terrain dominated by enemy air power." The Israelis had estimated some 1,000 tanks would be in the attack, so the fact that only 400 showed up was undoubtedly a relief.

The Israelis prepared for an armor battle, which some said they expected to be large and "savage" beyond belief. On the night of the 13 October, the Egyptians heli-lifted multiple commandos behind Israeli lines in the hopes of creating chaos in the Israeli rear. They were almost all captured and killed, quickly. The Israeli fortifications and preparations paid off, when the Egyptians launched the strike at dawn on the 14th. In the northern sector, the Egyptians were repelled with ease, and 50 tanks of theirs were destroyed. In the center, similar losses were had by the Egyptians. The Israelis had fortified on high ground, and fired on the charging Egyptian armor. Egypt's 1st Mechanized Brigade lost 93 tanks, effectively destroying the entire unit, and only 3 Israeli tanks in that area were destroyed: none by enemy tanks, all by rocket fire. In the south, near the Gidi and Mitla passes, the Egyptian attack was contained and the Israelis counterattacked, destroying some 60 Egyptian tanks. Egyptian forces tried to flank through Mitla pass to the south, and were stopped by the tanks they encountered as well as paratroopers. The battle ended with around 20 Israeli tanks destroyed. Egypt had lost around 260. The Egyptians had attacked superior gunnery, faced IAF superiority and bombing, hit fortified positions, and all with the sun in their eyes. In one swoop, Israeli forces watched the Egyptians retreat back to their bridgeheads on the East Bank of the Suez Canal, and an Egyptian general suffered a heart attack (Sa'ad Maamon) and had to be replaced by General Abd El Al Mona'am Wasel. All told, the Egyptians had taken such heavy losses that Israeli forces felt they could finally attempt a counterthrust, a genuine one that would cross the Suez. On 14 October, Israeli general Elazar gave orders for the crossing of the Suez the following night.

With regards to the Egyptian front, Israeli leaders were not only heartened by the sudden success, the airlift made them much more fluid and willing to fight. However, the Israelis were still fearful of a loss, and only in retrospect was the 14 October guaranteed as the turning point of the war in the Sinai. Israelis did not begin to have a confident assessment of their prospects until the 16 October, as the American resupply effort was already underway and helping replace equipment lost in the previous fighting. The airlift was not fully intended to "save" the Israelis, but to ensure they could continue their momentum. It was helpful that as soon as it began, Israel had begun to win on both fronts. Thus it was not so much that the airlift came when Israel was "close to defeat" and "saved", but rather it came when the Israelis thought they'd be defeated (it was ordered October 9, begun October 14, Kissinger blamed it on the Pentagon), and gave them more flexibility when they began a counterattack that might've failed because of the Israeli inability to win a war of attrition without getting crucial ammunition and resupply help.

On the Syrian front, the tide had begun to turn earlier, prompting the Syrian pleas to the Egyptians for relief. See, the Egyptians had promised the Syrians they would advance far further than they actually planned to into the Sinai. When the Syrians began to falter, they pointed the finger at the Egyptians, who had stopped advancing. The Egyptians decided to advance as a result, with the results I detailed above.

The Syrians had begun to falter by October 10. The IAF had begun to overcome the Syrian air defenses, partially due to a lack of more defense rockets. The Israelis had been relentless, launching airstrikes in almost suicidal fashion over the 7-8, and the Syrians lost their determination and began to move back. By that point the Israeli counterattack began to drive back the Syrians, and by October 9 the Syrian thrusting forces were effectively surrounded in the "Hushniye pocket". Both sides took severe losses as Hushniye turned into an "armor graveyard", but the Syrians came out worse. Israel began bombing military airfields belonging to the Syrians by October 8, and almost all of them were useless by the 14th. 8 Phantom F-4s, on October 9, managed to get to Damascus and bomb the General Staff and Air Force Headquarters buildings, catching the Syrians by surprise. Only one Phantom was shot down, and the morale blow was crushing. The second wave, another 8 Phantom F-4s, had been slated to hit the same buildings but couldn't get through thick butts safely, so they dropped their bombs on Hushniye's large tank concentrations, contributing greatly to the fight there. The Israelis got slightly overconfident, and on the 10th mounted an insufficiently manned and prepared attack: based on 1967 they had expected the Syrian army to be on the brink of collapse. The attack took serious losses and was called off. On October 11 the Israelis launched an attack aimed to get to Damascus, all the way to the Syrian capital, but they encountered heavy resistance. While the Syrians could not attack, their fortifications and defensive posturing was still very strong, and both sides were taking heavy losses as a result, especially since once again the Israelis sent an undersized force, not expecting such heavy resistance of two strong lines. The Syrians now had higher morale (knowing their capital was under threat), and their SAMs had begun to be effective again (more missiles gotten, closer ranges), curtailing Israeli effectiveness in launching the attack. Eventually these became less effective as Israeli pilots got better at handling the SAMs, and by October 14 when the airlift began they were running many sorties on the Syrian rear, and Israeli forces had managed to fight a difficult campaign that got them within 20 miles of Damascus. There, the ground offensive stopped, and Israeli forces managed to (with the help of the resupply) hold off counterattacks, and make one last achievement on the ground: the capture of Mount Hermon on October 21-22. Because of the huge alarm the Syrians faced with the mounting attempts on their capital, around October 11 Assad appealed to Sadat to advance as he had promised, hence the Sadat response conveyed to Shazly on October 12. The airlift helped the Israelis take Hermon at best, and hold off counterattacks on the 16th and 19th, but also did not "save" Israel: it permitted them greater freedom and kept them afloat during a war that might've turned against them if they hadn't gotten more ammunition and equipment.

Edit: Adding a portion on "Why the airlift, then?"

Well, the airlift came with a few motivations. One was to match, and surpass, the Soviet airlift that had begun on October 10 for the Egyptians and Syrians. It succeeded, in that regard. The Israelis had implicitly threatened to use nuclear weapons by some reports, and gone to a nuclear alert they were sure the US would notice, threatening nuclear war especially if they ran out of conventional weapons and could not sustain their losses (which, at the time, were quite massive). The US also hoped to gain from it, and Kissinger egged the Israelis on, telling them to advance against Syria. He said to the Israeli ambassador: "The IDF must attack [Syria] with all its strength, as if it had another 40 aircraft in hand, and not stint on ammunition or aircraft, because the United States will supply everything." On October 13 he told them to continue their attack, apparently because he knew the airlift was about to begin. The airlift undoubtedly helped Israelis feel secure in advancing.

As a another question about the war,why does the arab world considers it a victory?is their claim correct?