There were a number of reasons why the British First Airborne at Arnhem and ultimately Operation Market Garden failed. The most significant was the failure to capture Nijmegen Bridge the first day.
Major-General James Gavin’s 82nd Airborne Division was told to prioritized and hold Groesbeek Heights, which overlooked the Reichswald to the south-east of Nijmegen. It was thought the German 406th infantry was hidden there with armored support. (This was actually completely false, there was no German presence)
Their delayed advance on Nijmegen bridge (seven hours) allowed the German SS 10th Panzer division to advance north to Arnhem, dig in and reenforced the north side of the bridge, preventing its capture. It wasn’t until the third day of the operation when British XXX corps arrived with tanks, that Nijmegen was captured along side the 82nd Airborne.
The Germans also effectively blocked the road north delaying the Allied breakthrough another 2 days. By this point it was too late for the paratroopers in Arnhem. Elements of the 9th SS Panzer Division and 10th SS Panzer Division surrounded and decimated the British 1st Airborne Division. Of the ten thousand men who had landed to take Arnhem, fourteen hundred were killed and over six thousand captured; only twenty-four hundred paratroopers managed to safely cross to the south bank of the Rhine and withdraw.