The short version is that Market Garden was unsuccessful because it was unable to reinforce the airborne attacks quickly enough via a single highway.

The American parachute drops were fairly successful. With a few exceptions they were able to secure their objectives--bridges and crossroads from Eindhoven to Nijmegen that the British armor would need to pass over and through to reach the ultimate objective. However, in a few instances the Americans were insufficiently rapid in their assaults on bridges, which held up the advance. Arnhem and its bridge over the Rhine was the ultimate objective. The British airborne assault on that city was less successful, as German resistance was heavier than expected. Meanwhile, the British ground offensive that was to link up with the airborne forces was consistently behind schedule. Partly this was due to a relative lack of haste, but mostly it was due to the normal difficulties of an offensive. There was German resistance (particularly effective were antitank guns concealed off the road that were able to snarl the offensive by stopping a vehicle or two), traffic snarls caused by a need to shuffle men and equipment around in the convoy (such as the need to bring up pontoon bridges from the rear of the column), and other problems.

Weather also played a role. This delayed the entry of the Polish airborne forces as well as reinforcements for the Americans and British. Resupply via the air was also disrupted, as was air support for the operation overall. Intelligence failures (especially in communication and comprehension of existing intelligence) led to German forces in Arnhem being much more robust than anticipated. Add in a number of times when a particular Allied force made choices that either caused delays or did not take advantage of momentary opportunities and you have enough friction to cause the overall offensive to fail.

I used to be very critical of Market Garden. Now I think it was a gamble of an offensive, but that it didn't fail for any one reason. The Allied attacks were conducted with valor, but perhaps the need for speed and risk-taking was insufficiently emphasized. The German defense was spirited and especially so at Arnhem. German demolition work was enough to cause delays at critical junctures. The British ground offensive was perhaps insufficiently rapid, but logistical needs (for the pontoon bridges, for refueling, etc) complicated matters. Poor communication didn't help the Allied cause (insufficient radios for the airborne, a lack of understanding between the airborne units and ground units, a lack of all units being on the same page regarding the structure and pace of the offensive). Add in weather delays as well. All of this was friction that slowed down the Allied offensive enough to keep it from crossing the Rhine.

So what went wrong with Market Garden? Instead I might give the answer that too few things went just right.

After the fierce attritional fighting around Caen from D-Day to the end of July, Allied successes and rapid advances in August prompted considerable optimism throughout Allied command; Eisenhower's aide Captain Harry Butcher observed in his diary that Ike "felt for some days it had been obvious that our military force could advance almost at will, subject only to supply", and Major-General Sir John Kennedy made a note on 6th September that "... if we go on at the same pace as of late, we should be in Berlin by the 28th...".

Eisenhower favoured an advance on a broad front, but the unanticipated speed of progress caused severe logistical issues, with supply lines unable to keep pace. Montgomery had always favoured a single thrust, and proposed Market Garden to outflank the Siegfried Line with a "carpet" of airborne troops capturing bridges for XXX Corps to advance over. There was a strong desire at SHAEF to use the recently formed First Allied Airborne Army, with numerous airborne operations of varying scale planned since D-Day but always cancelled, and Eisenhower agreed (in something of a fudge) to Market Garden as part of the broad front, with Montgomery's 21st Army Group having "limited priority" for supplies.

Allied confidence, if not overconfidence, meant that despite its unprecedented scale there was barely a week from the decision to launch Market Garden on September 10th to the attack itself on September 17th; General Brereton, commander of the First Allied Airborne Army, announced in his briefing of the 10th that "major decisions arrived at now must stand - and these have to be made immediately". Resistance was expected to be light; Roy Urquhart, commander of the 1st Airborne Division, wrote "On 13 September, four days before the attack, corps blithely passed on the information that the Germans in Holland had few infantry reserves and a total armoured strength of not more than fifty to a hundred tanks. And even if there was some evidence that the Germans were reinforcing their river lines near Arnhem and Nijmegen, the soldiers manning these defences were said to be few and 'of low medical category'." In fact 2 SS Panzer Corps, consisting of 9 and 10 SS Panzer Divisions, was recuperating and re-equipping around Arnhem, and though understrength was still a veteran outfit and fearsome opposition for lightly equipped paratroops. Information about the Panzer Corps was available from Dutch resistance, aerial reconnaissance and Ultra decrypts, but warnings (for example from Brian Urquhart, intelligence officer of 1st Airborne Division) went unheeded. Ultra decrypts on September 15th even identified that the headquarters of Field Marshal Model, Commander in Chief of Army Group B, moved to Oosterbeek at the very heart of Market Garden, but the plans proceeded without change.

With the massive numbers of airborne troops involved (over 30,000) there weren't enough aircraft to deliver them all in a single lift, causing the air assault to be split over multiple days. Some of the landing zones, especially around Arnhem and Nijmegen, were far from ideal, many miles from the bridge objectives due to terrain and worries about enemy flak. The ground assault (the 'Garden' part of Market Garden) also faced unfavourable terrain, much of XXX Corps being funnelled down a single road with boggy ground either side.

Though the US 82nd and 101st Airborne Divisions seized most of the crucial bridges in the initial attacks, the road bridge at Nijmegen remained in German hands, and the British 1st Airborne Division encountered stiff German resistance at Arnhem and ended up fragmented with a small force dug in at one end of the bridge. Command and control of the 1st Airborne Division was hampered by difficulties with communication, subject of a paper by Major John W. Greenacre, Assessing the Reasons for Failure: 1st British Airborne Division Signal Communications during Operation ‘Market Garden’

The advance of XXX Corps was slower than planned due to the heavier than expected resistance, and poor weather disrupted the follow-up airlifts and further supply operations; the Polish 1st Parachute Brigade were finally dropped on September 21st after two days of delays. Nijmegen bridge was only taken 20th September after boats were brought up for the 82nd Airborne to make an assault crossing of the river, and XXX Corps failed to push on until the following day, by which time the British paratroops had finally been driven off Arnhem bridge.

In summary: "quite a lot" went wrong with Market Garden, primarily an overambitious plan that failed to take into account the German forces in the area.

Sources:

Major-General R E Urquhart, Arnhem

Peter Harclerode, Arnhem: A Tragedy of Errors

Ronald Lewin, Ultra Goes to War