Clause 1. The House of Representatives shall be composed of Members chosen every second Year by the People of the several States, and the Electors in each State shall have the Qualifications requisite for Electors of the most numerous Branch of the State Legislature.
Every two years, voters ("Electors") get a chance to cast ballots to determine who will represent them in the House of Representatives. The last bit about "Qualifications requisite" just means that each state must allow anyone who can legally vote in state elections also to vote for US Representative; the states aren't allowed to limit voting rights for US House elections to a small elite.
Clause 2. No Person shall be a Representative who shall not have attained to the age of twenty five Years, and been seven Years a Citizen of the United States, and who shall not, when elected, be an Inhabitant of that State in which he shall be chosen.
Here are the job requirements for serving in the House: You need to be 25 years old, you need to have been a citizen for at least seven years, and you need to live in the state you want to represent in Congress. That's it!
Clause 3. Representatives and direct Taxes shall be apportioned among the several States which may be included within this Union, according to their respective Numbers, which shall be determined by adding to the whole Number of free Persons, including those bound to Service for a Term of Years, and excluding Indians not taxed, three fifths of all other Persons. The actual Enumeration shall be made within three Years after the first Meeting of the Congress of the United States, and within every subsequent Term of ten Years, in such Manner as they shall by Law direct. The Number of Representatives shall not exceed one for every thirty Thousand, but each State shall have at Least one Representative; and until such enumeration shall be made, the State of New Hampshire shall be entitled to chuse three, Massachusetts eight, Rhode-Island and Providence Plantations one, Connecticut five, New-York six, New Jersey four, Pennsylvania eight, Delaware one, Maryland six, Virginia ten, North Carolina five, South Carolina five, and Georgia three.
Bad spellers of the world, untie! Er, unite! Notice that even our esteemed Founding Fathers had a unique way of spelling "chuse." Moving on to the important stuff… this clause simply establishes that representation in the House is apportioned on the basis of population; that is, the more populous the state, the more seats it gets in the House. As you can see, in the First Congress of 1789, that meant that Virginia got ten seats and Massachusetts got eight, while Rhode Island and Delaware got just one each. Today, California (the most populous state) has 53 congressional seats, while seven states (Alaska, Wyoming, Montana, North Dakota, South Dakota, Vermont, and Delaware) have only one each. This clause also requires that a Census be taken every ten years, to count the population in each state in order to ensure that each state's representation in Congress remains proportional to its population, even as state populations change over time. (In recent decades, for example, fast-growing southern and western states have gained seats, while many northeastern states have lost them.)
Finally, take a close look at the crossed-out bits of this clause. They're crossed out because later amendments to the Constitution repealed them or made them obsolete. The brief mention of "Direct taxes" in the first sentence long made it impossible for the government to use a tax system like the one we have today, since it required taxes to be charged in proportion to each state's population, rather than in proportion to each individual's income. Only the passage of the 16th Amendment in 1913 got rid of this requirement, making it possible for the government to create the modern income tax system. (Oh happy day!) The longer crossed-out sentence is a remnant of the infamous three-fifths compromise, which allowed slave states to count three-fifths of their slave populations for the purpose of congressional apportionment, even though those slaves obviously had no democratic voting rights. The Framers of the Constitution were careful not to use the words "slave" or "slavery" anywhere in the document, instead referring here to slaves euphemistically as "all other Persons." The passage of the 13th Amendment, which banned slavery following the Civil War, finally struck this offensive passage from the Constitution.
Clause 4. When vacancies happen in the Representation from any State, the Executive Authority thereof shall issue Writs of Election to fill such Vacancies.
If a congressional seat should become vacant in the middle of a term, for whatever reason, the state's governor (a.k.a. "the Executive Authority thereof") is supposed to call a special election to fill the vacancy.
Clause 5. The House of Representatives shall chuse their Speaker and other Officers; and shall have the sole Power of Impeachment.
For reasons that are not entirely clear, this clause includes two important elements that seemingly have nothing whatsoever to do with each other. First, the House of Representatives has the power to choose its own leaders; customarily, the majority party chooses its leader to serve as the Speaker of the House. Second, the House of Representatives—and only the House of Representatives—has the power to impeach executive and judicial officers deemed unfit for office. (To "impeach" means to accuse; if the House does vote to impeach, the Senate must then decide whether or not to convict and remove the person from office.)