Unlike our present President, I don't claim to be a constitutional scholar. However, the first sentence here is pretty straightforward. Based strictly on it, any state must recognize the marriages that are legal in every other state.
"Congress may by general laws prescribe the manner in which (they) shall be proved" is also pretty clear. It means Congress prescribes the methods by which judicial acts in one state, such as marriages, are to be documented for acceptance by another state.
"Congress may by general laws prescribe ... the effect thereof." is where the interpretation comes in. Given the Founders' notorious suspicion of federal power, I seriously doubt they intended to give Congress the power to exclude certain "public acts, records, and judicial proceedings" from the requirements of the first sentence whenever it so chose. This would give Congress, in essence, the power to decide which "public acts, records, and judicial proceedings" of the states need be accepted by other states. If that was their intent, why didn't they just say so?
I suspect the Defense of Marriage Act is unconstitutional for perfectly conservative and states' rights reasons.
Does Congress have, or should it have, the power to decide which state laws must be respected by other states? Or did the Constitution intend to give Congress the power to regulate how such laws would be communicated between states?