Q:

What was the contemporary popular opinion of the Anglo-Irish Treaty? 

After the Irish War of Independence (also known as The [Black and] Tan War), the Irish and British signed a treaty in 1921, which created the Irish Free State and started the process of gradual independence from British rule. 
The Irish Republican Army, which had been the primary military force on the Irish side, split in reaction to the treaty, because it failed to grant their demands:

Complete independence from Britain (the treaty granted only partial autonomy and self-determination of domestic affairs, but not total independence and sovereignty over international relations and self defense). 

An Irish Republic encompassing the entire island of Ireland (only 26 of the 32 counties of Ireland were included in the Free State). 

The total separation from the British crown (the treaty required Irish politicians to sign an oath of loyalty to the crown). 

As a result of the split within the ranks of the IRA, a disastrous civil war ensued, in which more lives were lost than had been lost in the war of independence.   The rival factions were labeled "Pro-treaty" (or "Treatyites") and "Anti-treaty" forces. 
Do we have any contemporary evidence of popular opinion in the Irish Free State regarding the treaty? 

A:

The best barometer we have of the attitude of the general population of the Irish Free State towards the Anglo-Irish Treaty are the "Pact Elections" of 16 June 1922.  They occurred twelve days before the commencement of hostilities in the Irish Civil War.  As the linked Wikipedia article points out, 75% percent of the electorate supported pro-Treaty parties.  We can also bear witness to the Irish electorate's overwhelming and uninterrupted support of Charles Stewart Parnell's Irish Parliamentary Party from the 1870s through WWI, during which time Parnell and his successor John Redmond advocated for a much more conservative form of Irish home rule than was stipulated in the Anglo-Irish Treaty.  
I think it's still within the scope of your question to discuss why you are asking it in the first place (i.e., Why, if the Irish electorate overwhelming supported the treaty, did civil war break out?)  In his excellent biography on Michael Collins, TP Coogan makes a convincing argument that the Irish Civil War was less a "civil war" than a conflict between highly factionalized professionals (i.e. opposing members of the Irish political establishment and their respective supporters in the Republican Army).  On the one side, you had Michael Collins and the other "compromisers" who genuinely believed that they had, through tireless negotiation with their haughty British overlords, finally delivered to Ireland the "freedom to achieve her freedom".  And on the other side, you had the majority of the Republican Army, whose members, for practical and logistical reasons, had been left out of the negotiating process and whose fiercely anti-British ideology prevented them from accepting anything less than total independence (which you point out in your question). 
Since both groups were heavily armed and seasoned veterans from years of conflict with the British, a violent clash of ideology vs. compromise was pretty much a foregone conclusion.  Yet according to Coogan, in this time period, everyday Irish citizens prioritized their confessional and economic freedom over unadulterated Irish nationalism (though this would shift later in the century as unification with North Ireland emerged as a major political cause celebre), hence, they did not feel drawn to engage in partisan violence. 
This summary is grossly over-simplified for the sake of space.  If you haven't already, I'd suggest you explore biographies of both Michael Collins (see TP Coogan's) and Eamon de Valera (a great one also written by Coogan).  Both of these men had an insane amount of influence on the process of shaping modern Ireland.  In fact, if it weren't for de Valera's public repudiation of the Treaty, the Republican Army would have had little pretext to do the same, and the Irish Civil War probably wouldn't have happened.