As you are likely aware, the invasion and occupation of Iceland was carried out with no meaningful resistance, but no country is particularly happy to be invaded and occupied against their will of course, and Iceland was no different. Still connected to Denmark via personal union with the King, Iceland was by 1940 a sovereign and independent monarchy insofar as its domestic matters were concerned, but Denmark continued to have say in foreign affairs. Although Iceland's government in theory had the power to direct how Denmark represented them on the world stage, it didn't really care all that much so left it to the Danes in practice.

Declaring neutrality at the beginning of the war, Iceland decidedly wanted nothing to do with the conflict, but with the fall of Denmark in April, 1940, the British were concerned that the Germans would be able to parlay control of Denmark into an arrangement to utilize Iceland as well. Britain made brief attempts to bring Iceland into the conflict as a belligerent, or at least as a cooperating non-belligerent, which would have of course solved that problem, but Iceland stood firm. Ironically, perhaps, they expected that British control of the sealanes was absolute enough that they already were firmly within the British sphere of influence. A cynic might say they wanted to have their cake and eat it too: being neutral and able to maintain diplomatic and trade relations on their own terms while nevertheless being protected from any German threats by the British. Whether the chance of German invasion was realistic or not (hint, it wasn't), Britain had been monitoring reports of German reconnaissance of Iceland since before the war in 1939, and wouldn't take that as a stance in any case and invaded a month after Denmark fell.

Again, there was no physical resistance, merely voicing of protest, and it must be said that even in that regard the Icelandic government was considerably more muted than one would expect in the face of an invasion, reflecting in large part British promises to maintain a light touch, abstaining from any say in domestic affairs and leaving as soon as the war ended. Allowed to address his country over the radio soon after, Hermann Jónasson, the Prime Minister, decried the "violation of Iceland’s neutrality and the impairment of its independence" but at the same time asked his countrymen "to treat the British soldiers [...] as guests and to show them, as any other guest, full courtesy in every way".

Generally speaking, the Iceland went back to normalcy quite quickly, and if anything, the inhabitants quickly came to be quite appreciative of their occupiers. Least of all they knew it was a better deal than if the Germans had showed up, but in more practical terms, the presence of British troops quickly proved to be an economic boon. Viewed now in hindsight, blessað stríðið, or 'blessed war', is a common way to refer to the war.

Germany of course protested, and while Hitler did express a desire to now invade Iceland, the Navy of course told him it wasn't gonna' happen, although they did soon after expand their operational area for U-Boat sorties to include the Icelandic region. Still unsure of whether Britain could actually protect them, Iceland also began a dialogue with the United States about their willingness to ensure Icelandic protection. Although initially unwilling to make any commitments, nevertheless British and American discussions over aid in early 1941 also began discussing the potential of the (still technically neutral) United States taking over Iceland to free up British forces for the war.

Unwilling to proceed with the same 'act first' mentality of the British though, as well as construing Iceland to be "Western Hemisphere" to fit under the Monroe Doctrine, the United States instead sought Icelandic invitation. The Althing was actually opposed to such a request, which presumably would further weaken claims of their continued neutrality, but Jónasson's government saw a way to bring some agreeable change and gain leverage. Eight conditions were laid out by Jónasson in a letter addressed to Roosevelt on June 24, 1941, which the United States agreed to, including:

1. United States promise to withdraw all their military forces land, air and sea from Iceland immediately on conclusion of present war.
2. United States further promise to recognize the absolute independence and sovereignty of Iceland and [and ensure the peace treaty has everyone else] recognize the absolute independence and sovereignty of Iceland.
3. United States promise not to interfere with Government of Iceland neither while their armed forces remain in this country nor afterwards.
4. United States promise to organize the defense of the country in such a way as to ensure the greatest possible safety for the inhabitants themselves and assure that they suffer minimum disturbance from military activities [...]
5. United States undertake defense of the country without expense to Iceland and promise compensation for all damage occasioned to the inhabitants by their military activities.
6. United States promise to further interests of Iceland in every way in their power, including that of supplying the country with sufficient necessities, of securing necessary shipping to and from the country and of making in other respects favorable commercial and trade agreements with it.
7. Iceland Government expects that declaration made by President in this connection will be in agreement with these promises on the part of Iceland, and Government would much appreciate its being given the opportunity of being cognizant with wording of this declaration before it is published.
8. On the part of Iceland it is considered obvious that if United States undertake defense of the country it must be strong enough to meet every eventuality and particularly in the beginning it is expected that as far as possible effort will be made to prevent any special danger in connection with change-over. [...]

Duly agreed to by Roosevelt, American Marines began to land on July 7th, 1941 to replace British forces, soon to be bolstered by the Army's 5th Infantry Division. Still not a belligerent, the United States portrayed their actions as defensive to ensure freedom of navigation and furthermore, with the invitation by the Icelandic government, to ensure the country's independence. As Jónasson had written out in his letter to Roosevelt just what he needed:

[T]he British Minister explained that British forces in Iceland are required elsewhere. At the same time he stressed the immense importance of adequate defense of Iceland. He also called my attention to the declaration of the President of the United States to the effect that he must take all necessary measures to ensure the safety of the Western Hemisphere-one of the President's measures is to assist in the defense of Iceland-and that the President is therefore prepared to send here immediately United States troops to supplement and eventually to replace the British force here. But that he does not consider that he can take this course except at the invitation of the Iceland Government.

After careful consideration of all the circumstances the Iceland Government, in view of the present state of affairs, admit that this measure is in accordance with the interest of Iceland, and therefore are ready to entrust the protection of Iceland to United States.

The United States mostly held up their end of the bargain, and it did help reduce some of the internal pressure in returning at least some sense of control to Iceland, however much of a pretense the invitation might seem to some. The only vocal opposition against the Allied presence came from the far left Socialist Unity Party, who declared the country:

had been turned into a target in the bloody conflict between the British and German capitalists fighting over world markets and resources – or into petty change in international negotiations between the capitalist superpowers to be used at the conclusion of the war.

Of course in the middle of 1941, they changed their tune considerably. As one subsequent article from the party cried out:

Your fate, your national liberty, your children’s future depends on the opponents of Fascism and the victory of the Red Army and other liberating military forces in this war. If they lose, then our freedom is lost, our culture destroyed.

With connections to the COMINTERN, this was of course an entirely principled reevaluation of what was moral and right, and in no way related to the Nazi invasion of the Soviet Union that summer.

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