The western allies had landed in northern France in Operation Overlord in June 1944 and then in southern France in Operation Dragoon and had encircled and routed the German counter-attack at Falaise in August 1944. After that, the western allies broke through and liberated most of France.
However, the Germans, while cut-off, still controlled most of the French ports and had supplies and fortifications enough to last in these strongholds for a long time - considering they also often had the ports mined or primed for demolition, they could deny them for the western allies even if twhey were ousted (at great cost). The wstern allies thus opted to levae most of these ports covered, but not make any serious attempts to capture them. Instead they focused on capturing the port of Antwerpen, one of Europe's largest, which was liberated on the 4th of September 1944. However, the Germans managed to keep the northern end of the Scheldt river, and without controlling it, Antwerpen was not safe for shipping, maning the western allies could not use the port to bring in much needed supplies.
The fighting in France, and the bombings carried out to prevent the Germans from reinforcing their counter-attack as well as German sabotage as they retreated had left the French infrastructure in shambles. This, combined with the temporary nature of the Mulberry harbours used in Normandy and the lengthening lines of supply meant that the western allied attack petered out in central Belgium and Alsace-Lorraine due to a general lack of supplies. Divisions use a lot of fuel and ammunition while in offensive operations, and enough could simply not be brought forwards.
Eisenhower favored wide-front attacks to overwhelm the Germans with allied superiority in air support, firepower and supply, but in this case it was not possible to conduct such operations, as there were not enough supply.
The British General Montgomery howevever, favored a limited deep surgical strike to cut through northern Belgium and southern Netherlands and capture a clear path, including brigdes, over the Rhine, allowing allied troops to cross the Rhine, cut of the German forces holding the northern bank of the Scheldt river, flank the defences in Alsace-Lorraine and pout into the vital Ruhr area, which provided a large percentage of German war industry and coal production, without which the country would not be able to continue the war.
Montgomery's plan called for the British 1. Airborne Division and the US 82. and 101. Airborne Divisions, with the Free Polish 1. Airlanding Brigade in reserve, to airdrop behind the German lines in Belgium and capture bridges at Arnhem, Nijmegen and OOsterbek while the British 30. Corps, veterans from the fighting in North Africa, would punch through the German lines and link up with the paratroopers and advance into Germany over the bridge at Arnhem.
The plan had several advantages - the paratroopers were avilable after being used in the initial landings in France to secure rear areas and would be supplied by air from Britain rather than over the devastated French infrastructure. The 30. Corps was a powerful and experienced mechanised formation, but still small enough that if supplies were prioritised, it could be used for offensive operations. Flanking the German defences in Alsace-Lorraine and the Siegfried line behind it and punching into Ruhr held a potential promise of ending the war around christimas time.
Several things went wrong, however. Either allied intelligence missed, or by miscommunication of willful ignorance (sources part on this) Montgomery and his staff missed that the Germans had deployed several depleted formations to rest and refit - which inckluded the II. SS Panzer-korps, which included two depleted but still strong SS Panzer-divisions.
The northern paratroopers had their landing zones and supply points over-run, and much of their airlanded supply ended up in German hands. The Germans reacted quickly, using their skill in improvisation to form several Kampfgruppen to fight both the paratroopers and the advancing 30. Corps from the various units available - which included elite formations resting and various cadre and training units, rear area and garrison forces and even units of wounded convalescents and cripples from ww1.
The allies had also over-estimated the ability of 30. Corps to advance - the terrain was soft and boggy, and autumn rains made it worse. The tanks, trucks and armoured personell carriers of 30. Corps were thus mostly stuck on elevated roads where they were easy targets for hidden German infantry equipped with Panzerschreck and Panzerfaust infantry anti-tank weapon or anti-tank guns, both of which together held up 30. Corps again and again.
In the end, the Germans were able to bring in more troops to the area than the allies were, and 30. Corps only managed to link up with the Americans and the Poles (dropped to reinforce the British at Arnhem, but ending up cut off) but the British 1. Airborne, or what was left of it, had to abandon the bridge at Arnhem and cross in the cover of darkness to escape the Germans - the operation had been as the famous quote 'a brigde too far'.
The western allies abandoned the idea of deep narrow front operations and Eisenhower's strategy of broad front advances became the norm. During the rest of autumn and early Winter, the allies cleared the northern bank of Scheldt river from German resistance and opened the port of Antwerpen to bring in supply - something which was a contributing factor in the Germans launching their offensive at the Battle of the Bulge, which intended to recapture Antwerpen or at least destroy it to deny its usage to the allies.
After pushing back the German offensive, reinforcing and resupplying and advancing their positions in Alsace-Lorraine and Belgium, the allies crossed the Rhine on a broad front in March 1945, surrounding and destroying the Ruhr in the process, and then raced across Germany to little resistance.