Why did Operation Market Garden go ahead when the allies knew German armor was in the area?

The What

Here are details on the incident from Wikipedia:

A number of reports about German troop movements reached Allied high command, including details about the identity and location of German armoured formations. Station X at Bletchley Park monitored and decrypted German Ultra intelligence reports and sent them to senior Allied commanders but they only reached army headquarters level and were not passed down any lower. On 16 September ULTRA decrypts revealed the movement of 9th and 10th SS Panzer Divisions to Nijmegen and Arnhem, creating enough concern for Eisenhower to send his Chief of Staff, Lieutenant General Walter Bedell Smith, to raise the issue with Montgomery on 10 September; however, Montgomery dismissed Smith's concerns and refused to alter the plans for the landing of 1st Airborne Division at Arnhem. Further information about the location of the German Panzer Divisions at Arnhem was revealed by aerial photographs of Arnhem taken by a photo-reconnaissance Spitfire XI from RAF's No. 16 Squadron, as well as information from members of the Dutch resistance.

Another source states that:

Major Urquhart recalled that five oblique angle pictures showed "the unmistakable presence of German armor" in the Arnhem area.

So, there was intelligence on the Nazi armoured divisions. The article goes on to say that:

Fearing that 1st Airborne Division might be in grave danger if it landed at Arnhem the chief intelligence officer of the division, Major Brian Urquhart, arranged a meeting with [Frederick] Browning and informed him of the armour present at Arnhem. Browning dismissed his claims and ordered the division's senior medical officer to send Urquhart on sick leave on account of 'nervous strain and exhaustion.'

So, Browning was the cause of this mistake. Here is another excerpt from Wikipedia.

Browning downplayed evidence brought to him by his intelligence officer, Major Brian Urquhart, that the 9th SS Panzer Division Hohenstaufen and the 10th SS Panzer Division Frundsberg were in the Arnhem area, but was not as confident as he led his subordinates to believe.

(emphasis mine)

The Why

There are a few possible reasons for this.

• Mistrust of the Dutch intelligence

British forces at Arnhem ignored the local Dutch resistance. There was a good reason for this: Britain's spy network in the Netherlands had been thoroughly and infamously compromised — the so-called England game, which had only been discovered in April 1944. Perhaps assuming that the Dutch resistance would be similarly penetrated, British intelligence took pains to minimise all civilian contact.

If the British had heeded word from their agents in Arnhem, they would have been alerted to the presence of two enemy panzer divisions.

• False assumptions

This one played the bigger role of the two. The allied commanders believed that the Germans were still retreating in an unorganized fashion and were incapable of putting up an effective resistance. This made them believe that the airborne corps would meet little resistance. Otherwise, they might have paid more heed to the reports of German armour. The allies correctly believed the Germans had about 50 tanks. What they did not take into account however, was that, "the soldiers crewing these tanks were some of the best in the Wehrmacht". Also, the panzer strength was growing daily, because of superb German logistics. The allies were able to put together a "fairly coherent picture of German equipment strength, as of 01 September." However, by the 17th of September it had completely changed. Intelligence analysts vastly underestimated the recuperative ability of the German logistical reinforcement system.

Other Sources

• The Dutch Resistance and the OSS--Central Intelligence Agency

• Intelligence-M-G -I highly recommend reading this one