Each House shall be the Judge of the Elections, Returns and Qualifications of its own Members, and a Majority of each shall constitute a Quorum to do Business; but a smaller Number may adjourn from day to day, and may be authorized to compel the Attendance of absent Members, in such Manner, and under such Penalties as each House may provide.
Comment: This provision gives to the House and Senate the power to determine whether an election of a member is valid, and whether the person elected meets the Constitutional requirements for service. This type of provision was common throughout England and America - e.g.,eight states had similar clauses in their constitutions. In any event, this clause was not litigated in court for almost 200 years until the House refused to seat Adam C. Powell because he had allegedly embezzled House funds and lied to the House. Powell sought a judicial declaration of the matter.
In Powell v. McCormack, the Court held that although Article I Section 5 provided that "Each House shall be the Judge of the...Qualifications of its own Members," Article 1 Section 2 specifies exactly three qualifications -- age, citizenship, and residency -- and that implies that there can be no more qualifications added by Congress. Moreover, in Morgan v. United States, Judge Scalia held that the House's determination as to which candidates had been elected was not within the court's power to review (i.e. the issue was nonjusticiable).
Regarding the Quorum provision, there was a two-thirds requirement under the Articles of Confederation, which was a major onstacle that they did not want anymore. As a result, the Framers settled on a majority - and some even thought that was too high as well. The day-to-day adjournment provision allows the business of the Congress to carryon with a majority needing to be present. As a result, in todays' Congress a quorum is not usually considered, unless a member requests a quorum.
A real threat to the quorum requirement exists if a sufficient number of members decided to not attend so as to prevent a quorum and keep business from being done - e.g., Wisconsin. As a result, the Framers saw fit, thankfully, to permit each house "to compel the attendance of absent members," which passed without debate. Currently, fifteen Member of the House may order the Sergeant at Arms to compel the attendence of the absent Members.
Primary Source - The Heritage Guide to the Constitution.