If God infallibly knew, from before you were born, exactly what choice you would make at every moment of your life, then you could never have done otherwise than what God foreknew — which means human free will is an illusion, undermining both moral responsibility and the Christian doctrine that people freely choose to accept or reject God.

Why people find it persuasive

It seems like simple logic: if it was already true a thousand years ago that you would choose X tomorrow, then tomorrow you either choose X (in which case you couldn't have chosen otherwise without making God's past knowledge false, which is impossible) or the whole notion of infallible foreknowledge has to be given up — and giving up genuine alternative possibilities looks like giving up libertarian free will.

Important distinctions

• • Foreknowledge as causing an event vs. merely knowing what a free agent will (of their own accord) do.
• • Necessity of the consequent ('necessarily, if God foreknew X, X happens') vs. necessity of the consequence being illegitimately transferred to the antecedent event itself ('X happens necessarily').
• • Foreknowledge from within time (temporal, sequential knowledge of the future) vs. God's knowledge from a timeless, eternal vantage encompassing all of history as a single 'present'.
• • Compatibilist vs. libertarian conceptions of the freedom moral responsibility actually requires.

Christian response

• • The standard scholastic (Boethian) resolution distinguishes the necessity of the conditional from the necessity of what's conditioned: it is necessarily true that 'if God foreknows you will choose X, you will choose X' (this is just the definition of infallible knowledge — you can't know a falsehood), but it does not follow that your choosing X is itself necessary in the sense that matters for freedom (i.e., that you lack the genuine power to do otherwise); this is the classic distinction between the 'necessity of the consequence' (which is unobjectionable and applies to any correct knowledge of any true proposition, foreknowledge or not) and an illegitimate slide into the 'necessity of the consequent' (treating the foreknown event itself as unconditionally necessary), a modal fallacy that has been carefully diagnosed since Boethius and Aquinas.
• • Many classical theologians (Boethius, Aquinas, and much of the tradition following them) further resolve the puzzle by denying that God's knowledge is properly called 'foreknowledge' at all in the relevant temporally-prior sense — God, existing outside of time (eternally, in a timeless 'now' that encompasses all of temporal history as a single simultaneous whole from his perspective, an idea developed at length in Boethius's Consolation of Philosophy), does not know your Tuesday choice 'in advance' the way a fortune-teller predicts a future event from an earlier vantage; God knows your free choice as you make it, from his atemporal perspective, which is analogous to how someone watching an entire parade from a high tower sees every float 'at once' without any float's position being thereby caused by, or made unfree by, the observer's vantage.
• • An alternative, equally serious response within the Christian tradition (Molinism, associated with the 16th-century Jesuit Luis de Molina and defended by contemporary philosophers like William Lane Craig and Alvin Plantinga) holds that God has 'middle knowledge' — infallible knowledge of what every possible free creature would freely choose in every possible circumstance, logically prior to (though not caused by) his decision to create — allowing God to know and providentially order history around genuinely free creaturely choices without those choices being determined by God's knowledge of them, since the knowledge itself is grounded in (explanatorily dependent on) what the free creature would in fact freely do, not the reverse.
• • A third serious option (open theism, a minority but coherent position within the Christian tradition, defended by philosophers like William Hasker and Greg Boyd) simply denies that God has exhaustive foreknowledge of future free choices at all — holding that genuinely free future choices are not yet true or false propositions available to be known even by an omniscient being (a particular view of the logic of future contingents going back to Aristotle's sea-battle problem), preserving libertarian freedom straightforwardly at the cost of a less traditional account of omniscience, which most classical theologians reject as unnecessarily concessive but which shows the puzzle, even if judged intractable on other models, is not a forced problem for theism as such.
• • Any of these three serious, independently-motivated solutions (Boethian eternalism, Molinism, open theism) is sufficient to show the objection does not constitute a knock-down logical disproof of the compatibility of infallible divine knowledge and human freedom, even if reasonable philosophers continue to disagree about which of the three is correct, or about which conception of freedom (libertarian vs. compatibilist) is ultimately required for moral responsibility in the first place — a further live debate in general philosophy of action entirely independent of theology.

Christian counterresponse

These are exactly the technical difficulties actively debated in contemporary analytic philosophy of religion (the grounding objection to Molinism has generated an extensive technical literature with serious responses on both sides, e.g., from Thomas Flint and Alfred Freddoso defending Molinism against it), and their unresolved status cuts against overconfidence in either direction — it shows the foreknowledge/freedom problem is a genuinely difficult, actively worked technical puzzle in modal logic and metaphysics comparable to other hard problems in general (non-theological) philosophy of time and freedom, not a simple, decisive refutation of theism that the objection's confident, one-paragraph version suggests; philosophers on both sides of the theism question take all three of these positions seriously as live options.